Explanation:
Okay, so I need to find the length of DE in this diagram. Let me try to visualize it first. There's a right triangle DEF, right? Because there's a right angle mark at E, between D, E, and F. The Asymptote code shows that F is at (0,0), D is at (sqrt(51),7), and E is at (0,7). So, the triangle is drawn from D to E to F to D.

First, let me recall that in a right triangle, the sine of an angle is opposite over hypotenuse. The problem says that sin D = 0.7. So angle at D? Wait, let me confirm. The right angle is at E, so angles at D and F are acute angles. So angle D is at point D, which is connected to E and F.

So, sin D = 0.7. So, if I can figure out which sides correspond to the opposite and hypotenuse for angle D, I can set up an equation.

In triangle DEF, angle D is at point D. The sides relative to angle D are: the opposite side would be the side opposite angle D, which would be EF. The hypotenuse is DF, since that's the side opposite the right angle. Wait, the hypotenuse is DF? Because the right angle is at E, so the hypotenuse is the side opposite the right angle, which is DF. Therefore, sin D = opposite/hypotenuse = EF / DF = 0.7.

But I don't know EF or DF yet. However, maybe I can figure out some coordinates? Wait, the coordinates are given in the Asymptote code. Let me check that again.

F is at (0,0), D is at (sqrt(51),7), and E is at (0,7). So, coordinates:

- Point F: (0,0)
- Point E: (0,7)
- Point D: (sqrt(51),7)

So, plotting these points, E is straight up along the y-axis from F, since E is at (0,7) and F is at (0,0). Then D is somewhere to the right of E, since its x-coordinate is sqrt(51) and y-coordinate is 7. So, the triangle DEF connects D to E to F to D.

So, maybe I can calculate DE using coordinates? If I can find the coordinates of D and E, I can compute the distance between them. Wait, but E is at (0,7) and D is at (sqrt(51),7). So, both points have the same y-coordinate, 7. Therefore, the distance between D and E is just the difference in their x-coordinates.

So DE would be sqrt( (sqrt(51) - 0)^2 + (7 - 7)^2 ) = sqrt( (sqrt(51))^2 + 0 ) = sqrt(51). Wait, so DE is sqrt(51)? But that seems too straightforward. However, the problem says that sin D = 0.7. So maybe they want me to verify that or perhaps there's some confusion here.

Wait, maybe I need to check if my coordinates are consistent with sin D = 0.7? Let me verify that.

First, let's compute sin D. In triangle DEF, angle at D. The sides are:

- Opposite to angle D: EF
- Adjacent to angle D: DE
- Hypotenuse: DF

So, sin D = EF / DF = 0.7

First, let me compute EF. EF is from E(0,7) to F(0,0). The distance is sqrt( (0 - 0)^2 + (0 - 7)^2 ) = 7. So EF is 7.

Then DF is from D(sqrt(51),7) to F(0,0). The distance is sqrt( (sqrt(51) - 0)^2 + (7 - 0)^2 ) = sqrt(51 + 49) = sqrt(100) = 10.

Therefore, sin D = EF / DF = 7 / 10 = 0.7. Oh! So that's exactly the given value. Therefore, DE is sqrt(51) as calculated earlier. But the problem is asking for DE. Wait, but sqrt(51) is approximately 7.14, but maybe they want an exact value?

But let me check once more. The coordinates of D are (sqrt(51),7) and E is (0,7). So, since they have the same y-coordinate, the vertical component between D and E is zero, so DE is horizontal distance, which is sqrt(51) - 0 = sqrt(51). Therefore, DE is sqrt(51).

But the problem could be tricking me? Because maybe I need to compute it using trigonometry instead of coordinates? Let me try that approach as well to confirm.

So, sin D = 0.7 = 7/10. Therefore, in triangle DEF, opposite side EF = 7, hypotenuse DF = 10. Then, using Pythagoras, the adjacent side DE would be sqrt(DF² - EF²) = sqrt(10² - 7²) = sqrt(100 - 49) = sqrt(51). So DE is sqrt(51), same as before.

Therefore, regardless of the method, coordinates or trigonometry, DE is sqrt(51). Therefore, is that the answer?

But wait, let me check the diagram again. The Asymptote code shows labels: the segment between E and F has a label "7" in the middle, which is the length of EF, which we found is 7. So that's consistent with EF being 7. So, then DE is sqrt(51), and DF is 10.

But maybe they want it simplified or something else? Wait, sqrt(51) can't be simplified further because 51 factors into 3*17, neither of which are perfect squares. So sqrt(51) is simplest radical form.

Alternatively, if they wanted a decimal? But the problem gives sin D as 0.7, which is exact. So probably expects an exact answer.

But let me check if there's any possibility I misread the question. The problem says "What is DE?" with the diagram. The Asymptote code defines D at (sqrt(51),7), E at (0,7). Therefore, DE is sqrt( (sqrt(51))^2 + 0 ) = sqrt(51). So that's correct.

Alternatively, maybe the question is designed to have you use the sine value to find DE? But since we already have coordinates, maybe it's just a verification. But the problem says "in the diagram below" so maybe they expect using coordinates? But the coordinates are given in the Asymptote code. However, the problem might be presented without coordinates, and the Asymptote is just for illustration. Wait, the problem says "in the diagram below" but since the user hasn't provided an image, the Asymptote code is given.

But the problem says "Suppose sin D = 0.7 in the diagram below." So maybe the coordinates are not given in the actual problem, but in the Asymptote code? Wait, but the Asymptote is part of the problem. So in the original problem, maybe the diagram is as per the Asymptote code, which defines F at (0,0), D at (sqrt(51),7), E at (0,7).

But if that's the case, then DE is sqrt(51). However, maybe the problem is presented without coordinates, and the Asymptote is just to show the diagram. So perhaps the problem is expecting to compute DE using the given sin D = 0.7, without relying on coordinates? Wait, but the coordinates define the positions, so if you have coordinates, you can compute DE directly. However, maybe the problem is designed to test understanding of sine in a right triangle.

But if it's a right triangle with right angle at E, then angle D has opposite side EF = 7 (since EF is opposite angle D) and hypotenuse DF. Then, sin D = 7 / DF = 0.7. So DF = 7 / 0.7 = 10. Then, DE is the adjacent side, so DE = sqrt(DF² - EF²) = sqrt(100 - 49) = sqrt(51). Hence, DE = sqrt(51).

Therefore, regardless of the approach, the answer is sqrt(51).

But let me make sure that in the diagram, EF is indeed opposite angle D. Since angle D is at point D, which is connected to E and F. So, in triangle DEF, angle D: the sides are DE, DF, and EF. The side opposite angle D is EF, as EF is opposite to D. So yes, that's correct.

Alternatively, if someone thought that the side adjacent was EF, but no. In angle D, the adjacent side is DE, and the opposite is EF.

Therefore, everything checks out. So DE is sqrt(51).

But just to make sure, let me check with coordinate geometry. Coordinates of D: (sqrt(51), 7), coordinates of E: (0, 7). Then, DE is horizontal line, difference in x-coordinates is sqrt(51) - 0 = sqrt(51), difference in y-coordinates is 0. So distance is sqrt( (sqrt(51))² + 0² ) = sqrt(51). Yes.

Alternatively, using vectors: vector DE is E - D? Wait, no. From D to E is E - D. Coordinates of E minus coordinates of D: (0 - sqrt(51), 7 - 7) = (-sqrt(51), 0). The magnitude is sqrt( ( -sqrt(51) )² + 0² ) = sqrt(51).

Therefore, all methods confirm that DE is sqrt(51). Therefore, the answer is sqrt(51).

Wait, but maybe there is a trick here? The problem says "in the diagram below" but perhaps in the actual diagram, the coordinates are different? Wait, but since the Asymptote code is given, and that defines the coordinates, then the answer must be sqrt(51). However, let me check once again if the Asymptote code is accurate.

Looking back, the Asymptote code defines:

F = (0,0);
D = (sqrt(51),7);
E = (0,7);
draw(D--E--F--D);
draw(rightanglemark(D,E,F,15));
label("$D$",D,NE);
label("$E$",E,NW);
label("$F$",F,SW);
label("$7$",(E+F)/2,W);

So, the segment EF is from E(0,7) to F(0,0), which is vertical line, labeled 7, as the midpoint between E and F is (0, 3.5), and the label is placed to the west (left), so that's correct. Then the right angle is at E, between D-E-F. So, angle at E is the right angle. So triangle DEF is right-angled at E, with legs DE and EF, and hypotenuse DF. So EF is 7, DE is sqrt(51), DF is 10. Then sin D is opposite over hypotenuse = EF / DF = 7/10 = 0.7, which matches the given information. Therefore, the diagram is consistent with the given sin D = 0.7. Therefore, DE is sqrt(51).

Therefore, I think the answer is sqrt(51).

But let me check if the problem is from somewhere? Wait, no, the user just provided this problem with Asymptote code.

Alternatively, maybe there's a miscalculation? Let me check the DF length again. Coordinates of D: (sqrt(51),7), coordinates of F: (0,0). Then DF distance is sqrt( (sqrt(51) - 0)^2 + (7 - 0)^2 ) = sqrt(51 + 49) = sqrt(100) = 10. Yes. So DF = 10. Then EF is 7. Therefore, DE = sqrt(DF² - EF²) = sqrt(100 - 49) = sqrt(51).

Therefore, no problem here.

Alternatively, if someone thought that angle D is at the right angle? But no, the right angle is at E. So angle D is one of the acute angles.

So I think the answer is sqrt(51). Therefore, the answer is \boxed{\sqrt{51}}.

**Final Answer**
\boxed{\sqrt{51}}

To determine the length of segment $ DE $, we are given a right triangle $ \triangle DEF $ with a right angle at $ E $, and that $ \sin D = 0.7 $. Also, the coordinates of the triangle's vertices are:
- $ F = (0, 0) $
- $ E = (0, 7) $
- $ D = (\sqrt{51}, 7) $

---

### Step 1: Understand the Triangle and Angle

Since the right angle is at $ E $, the triangle is right-angled at $ E $. Therefore:
- $ \angle D $ is one of the acute angles.
- The side opposite to $ \angle D $ is $ EF $.
- The hypotenuse of the triangle is $ DF $.

Given:
$$
\sin D = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EF}{DF} = 0.7
$$

---

### Step 2: Compute Side Lengths from Coordinates

Using the coordinates of points:
- $ E = (0, 7) $
- $ F = (0, 0) $
- $ D = (\sqrt{51}, 7) $

We can compute the lengths directly:

#### Side $ EF $:
$$
EF = \sqrt{(0 - 0)^2 + (0 - 7)^2} = \sqrt{0 + 49} = 7
$$

#### Side $ DF $:
$$
DF = \sqrt{(\sqrt{51} - 0)^2 + (7 - 0)^2} = \sqrt{51 + 49} = \sqrt{100} = 10
$$

#### Side $ DE $:
Since both $ D $ and $ E $ have the same $ y $-coordinate (7), the segment $ DE $ lies horizontally, and thus:
$$
DE = \text{horizontal distance} = |\sqrt{51} - 0| = \sqrt{51}
$$

---

### Step 3: Verification Through Trigonometry

From the definition of sine:
$$
\sin D = \frac{EF}{DF} = \frac{7}{10} = 0.7
$$
This matches the given value.

To confirm the length of $ DE $, we can use the Pythagorean Theorem:
$$
DE = \sqrt{DF^2 - EF^2} = \sqrt{10^2 - 7^2} = \sqrt{100 - 49} = \sqrt{51}
$$

---

### Final Result

All approaches—coordinate geometry, vector distance, and trigonometry—lead to the same result:

$$
\boxed{\sqrt{51}}
$$

[Assistant]
Explanation:
Okay, so I need to find the length of DE in this diagram. Let me try to visualize it first. There's a right triangle DEF, right? Because there's a right angle mark at E, between D, E, and F. The Asymptote code shows that F is at (0,0), D is at (sqrt(51),7), and E is at (0,7). So, the triangle is drawn from D to E to F to D.

First, let me recall that in a right triangle, the sine of an angle is opposite over hypotenuse. The problem says that sin D = 0.7. So angle at D? Wait, let me confirm. The right angle is at E, so angles at D and F are acute angles. So angle D is at point D, which is connected to E and F.

So, sin D = 0.7. So, if I can figure out which sides correspond to the opposite and hypotenuse for angle D, I can set up an equation.

In triangle DEF, angle D is at point D. The sides relative to angle D are: the opposite side would be the side opposite angle D, which would be EF. The hypotenuse is DF, since that's the side opposite the right angle. Wait, the hypotenuse is DF? Because the right angle is at E, so the hypotenuse is the side opposite the right angle, which is DF. Therefore, sin D = opposite/hypotenuse = EF / DF = 0.7.

But I don't know EF or DF yet. However, maybe I can figure out some coordinates? Wait, the coordinates are given in the Asymptote code. Let me check that again.

F is at (0,0), D is at (sqrt(51),7), and E is at (0,7). So, coordinates:

- Point F: (0,0)
- Point E: (0,7)
- Point D: (sqrt(51),7)

So, plotting these points, E is straight up along the y-axis from F, since E is at (0,7) and F is at (0,0). Then D is somewhere to the right of E, since its x-coordinate is sqrt(51) and y-coordinate is 7. So, the triangle DEF connects D to E to F to D.

So, maybe I can calculate DE using coordinates? If I can find the coordinates of D and E, I can compute the distance between them. Wait, but E is at (0,7) and D is at (sqrt(51),7). So, both points have the same y-coordinate, 7. Therefore, the distance between D and E is just the difference in their x-coordinates.

So DE would be sqrt( (sqrt(51) - 0)^2 + (7 - 7)^2 ) = sqrt( (sqrt(51))^2 + 0 ) = sqrt(51). Wait, so DE is sqrt(51)? But that seems too straightforward. However, the problem says that sin D = 0.7. So maybe they want me to verify that or perhaps there's some confusion here.

Wait, maybe I need to check if my coordinates are consistent with sin D = 0.7? Let me verify that.

First, let's compute sin D. In triangle DEF, angle at D. The sides are:

- Opposite to angle D: EF
- Adjacent to angle D: DE
- Hypotenuse: DF

So, sin D = EF / DF = 0.7

First, let me compute EF. EF is from E(0,7) to F(0,0). The distance is sqrt( (0 - 0)^2 + (0 - 7)^2 ) = 7. So EF is 7.

Then DF is from D(sqrt(51),7) to F(0,0). The distance is sqrt( (sqrt(51) - 0)^2 + (7 - 0)^2 ) = sqrt(51 + 49) = sqrt(100) = 10.

Therefore, sin D = EF / DF = 7 / 10 = 0.7. Oh! So that's exactly the given value. Therefore, DE is sqrt(51) as calculated earlier. But the problem is asking for DE. Wait, but sqrt(51) is approximately 7.14, but maybe they want an exact value?

But let me check once more. The coordinates of D are (sqrt(51),7) and E is (0,7). So, since they have the same y-coordinate, the vertical component between D and E is zero, so DE is horizontal distance, which is sqrt(51) - 0 = sqrt(51). Therefore, DE is sqrt(51).

But the problem could be tricking me? Because maybe I need to compute it using trigonometry instead of coordinates? Let me try that approach as well to confirm.

So, sin D = 0.7 = 7/10. Therefore, in triangle DEF, opposite side EF = 7, hypotenuse DF = 10. Then, using Pythagoras, the adjacent side DE would be sqrt(DF² - EF²) = sqrt(10² - 7²) = sqrt(100 - 49) = sqrt(51). So DE is sqrt(51), same as before.

Therefore, regardless of the method, coordinates or trigonometry, DE is sqrt(51). Therefore, is that the answer?

But wait, let me check the diagram again. The Asymptote code shows labels: the segment between E and F has a label "7" in the middle, which is the length of EF, which we found is 7. So that's consistent with EF being 7. So, then DE is sqrt(51), and DF is 10.

But maybe they want it simplified or something else? Wait, sqrt(51) can't be simplified further because 51 factors into 3*17, neither of which are perfect squares. So sqrt(51) is simplest radical form.

Alternatively, if they wanted a decimal? But the problem gives sin D as 0.7, which is exact. So probably expects an exact answer.

But let me check if there's any possibility I misread the question. The problem says "What is DE?" with the diagram. The Asymptote code defines D at (sqrt(51),7), E at (0,7). Therefore, DE is sqrt( (sqrt(51))^2 + 0 ) = sqrt(51). So that's correct.

Alternatively, maybe the question is designed to have you use the sine value to find DE? But since we already have coordinates, maybe it's just a verification. But the problem says "in the diagram below" so maybe they expect using coordinates? But the coordinates are given in the Asymptote code. However, the problem might be presented without coordinates, and the Asymptote is just for illustration. Wait, the problem says "in the diagram below" but since the user hasn't provided an image, the Asymptote code is given.

But the problem says "Suppose sin D = 0.7 in the diagram below." So maybe the coordinates are not given in the actual problem, but in the Asymptote code? Wait, but the Asymptote is part of the problem. So in the original problem, maybe the diagram is as per the Asymptote code, which defines F at (0,0), D at (sqrt(51),7), E at (0,7).

But if that's the case, then DE is sqrt(51). However, maybe the problem is presented without coordinates, and the Asymptote is just to show the diagram. So perhaps the problem is expecting to compute DE using the given sin D = 0.7, without relying on coordinates? Wait, but the coordinates define the positions, so if you have coordinates, you can compute DE directly. However, maybe the problem is designed to test understanding of sine in a right triangle.

But if it's a right triangle with right angle at E, then angle D has opposite side EF = 7 (since EF is opposite angle D) and hypotenuse DF. Then, sin D = 7 / DF = 0.7. So DF = 7 / 0.7 = 10. Then, DE is the adjacent side, so DE = sqrt(DF² - EF²) = sqrt(100 - 49) = sqrt(51). Hence, DE = sqrt(51).

Therefore, regardless of the approach, the answer is sqrt(51).

But let me make sure that in the diagram, EF is indeed opposite angle D. Since angle D is at point D, which is connected to E and F. So, in triangle DEF, angle D: the sides are DE, DF, and EF. The side opposite angle D is EF, as EF is opposite to D. So yes, that's correct.

Alternatively, if someone thought that the side adjacent was EF, but no. In angle D, the adjacent side is DE, and the opposite is EF.

Therefore, everything checks out. So DE is sqrt(51).

But just to make sure, let me check with coordinate geometry. Coordinates of D: (sqrt(51), 7), coordinates of E: (0, 7). Then, DE is horizontal line, difference in x-coordinates is sqrt(51) - 0 = sqrt(51), difference in y-coordinates is 0. So distance is sqrt( (sqrt(51))² + 0² ) = sqrt(51). Yes.

Alternatively, using vectors: vector DE is E - D? Wait, no. From D to E is E - D. Coordinates of E minus coordinates of D: (0 - sqrt(51), 7 - 7) = (-sqrt(51), 0). The magnitude is sqrt( ( -sqrt(51) )² + 0² ) = sqrt(51).

Therefore, all methods confirm that DE is sqrt(51). Therefore, the answer is sqrt(51).

Wait, but maybe there is a trick here? The problem says "in the diagram below" but perhaps in the actual diagram, the coordinates are different? Wait, but since the Asymptote code is given, and that defines the coordinates, then the answer must be sqrt(51). However, let me check once again if the Asymptote code is accurate.

Looking back, the Asymptote code defines:

F = (0,0);
D = (sqrt(51),7);
E = (0,7);
draw(D--E--F--D);
draw(rightanglemark(D,E,F,15));
label("$D$",D,NE);
label("$E$",E,NW);
label("$F$",F,SW);
label("$7$",(E+F)/2,W);

So, the segment EF is from E(0,7) to F(0,0), which is vertical line, labeled 7, as the midpoint between E and F is (0, 3.5), and the label is placed to the west (left), so that's correct. Then the right angle is at E, between D-E-F. So, angle at E is the right angle. So triangle DEF is right-angled at E, with legs DE and EF, and hypotenuse DF. So EF is 7, DE is sqrt(51), DF is 10. Then sin D is opposite over hypotenuse = EF / DF = 7/10 = 0.7, which matches the given information. Therefore, the diagram is consistent with the given sin D = 0.7. Therefore, DE is sqrt(51).

Therefore, I think the answer is sqrt(51).

But let me check if the problem is from somewhere? Wait, no, the user just provided this problem with Asymptote code.

Alternatively, maybe there's a miscalculation? Let me check the DF length again. Coordinates of D: (sqrt(51),7), coordinates of F: (0,0). Then DF distance is sqrt( (sqrt(51) - 0)^2 + (7 - 0)^2 ) = sqrt(51 + 49) = sqrt(100) = 10. Yes. So DF = 10. Then EF is 7. Therefore, DE = sqrt(DF² - EF²) = sqrt(100 - 49) = sqrt(51).

Therefore, no problem here.

Alternatively, if someone thought that angle D is at the right angle? But no, the right angle is at E. So angle D is one of the acute angles.

So I think the answer is sqrt(51). Therefore, the answer is \boxed{\sqrt{51}}.

**Final Answer**
\boxed{\sqrt{51}}

To determine the length of segment $ DE $, we are given a right triangle $ \triangle DEF $ with a right angle at $ E $, and that $ \sin D = 0.7 $. Also, the coordinates of the triangle's vertices are:
- $ F = (0, 0) $
- $ E = (0, 7) $
- $ D = (\sqrt{51}, 7) $

---

### Step 1: Understand the Triangle and Angle

Since the right angle is at $ E $, the triangle is right-angled at $ E $. Therefore:
- $ \angle D $ is one of the acute angles.
- The side opposite to $ \angle D $ is $ EF $.
- The hypotenuse of the triangle is $ DF $.

Given:
$$
\sin D = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EF}{DF} = 0.7
$$

---

### Step 2: Compute Side Lengths from Coordinates

Using the coordinates of points:
- $ E = (0, 7) $
- $ F = (0, 0) $
- $ D = (\sqrt{51}, 7) $

We can compute the lengths directly:

#### Side $ EF $:
$$
EF = \sqrt{(0 - 0)^2 + (0 - 7)^2} = \sqrt{0 + 49} = 7
$$

#### Side $ DF $:
$$
DF = \sqrt{(\sqrt{51} - 0)^2 + (7 - 0)^2} = \sqrt{51 + 49} = \sqrt{100} = 10
$$

#### Side $ DE $:
Since both $ D $ and $ E $ have the same $ y $-coordinate (7), the segment $ DE $ lies horizontally, and thus:
$$
DE = \text{horizontal distance} = |\sqrt{51} - 0| = \sqrt{51}
$$

---

### Step 3: Verification Through Trigonometry

From the definition of sine:
$$
\sin D = \frac{EF}{DF} = \frac{7}{10} = 0.7
$$
This matches the given value.

To confirm the length of $ DE $, we can use the Pythagorean Theorem:
$$
DE = \sqrt{DF^2 - EF^2} = \sqrt{10^2 - 7^2} = \sqrt{100 - 49} = \sqrt{51}
$$

---

### Final Result

All approaches—coordinate geometry, vector distance, and trigonometry—lead to the same result:

$$
\boxed{\sqrt{51}}
$$

Solution:
The triangle is a right triangle, so $\sin D = \frac{EF}{DF}$. Then we have that $\sin D = 0.7 = \frac{7}{DF}$, so $DF = 10$.

Using the Pythagorean Theorem, we find that the length of $DE$ is $\sqrt{DF^2 - EF^2},$ or $\sqrt{100 - 49} = \boxed{\sqrt{51}}$.


Explanation:
Okay, so I need to compute this series: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Let me think about how to approach this.

First, I notice that the series alternates between adding and subtracting numbers. It starts with 1, then subtracts 2, adds 3, subtracts 4, and so on, up to 100. Since the last term is -100, that means there are 100 terms in total. Let me check that. The first term is 1, then each pair of terms is like (1 - 2), (3 - 4), ..., up to (99 - 100). So if I group them in pairs, how many pairs are there?

Well, since there are 100 terms, and each pair has 2 terms, there should be 100 / 2 = 50 pairs. Each pair is of the form (odd number - even number). Let me write out the first few pairs to see the pattern:

(1 - 2) + (3 - 4) + (5 - 6) + ... + (99 - 100)

Each of these pairs simplifies to -1. Let me verify that:

1 - 2 = -1

3 - 4 = -1

5 - 6 = -1

...

99 - 100 = -1

So each pair is indeed -1. Since there are 50 pairs, the total sum would be 50 * (-1) = -50.

Wait, that seems straightforward, but let me make sure I haven't missed anything. Let me check with a smaller example. Let's say I have 1 - 2 + 3 - 4. According to my method, there are 4 terms, so 2 pairs. Each pair is -1, so total is -2. Let me compute manually: 1 - 2 + 3 - 4 = (1 - 2) + (3 - 4) = (-1) + (-1) = -2. That's correct.

Another example: 1 - 2 + 3 - 4 + 5 - 6. That's 3 pairs, so -3. Calculating manually: 1 - 2 = -1; -1 + 3 = 2; 2 - 4 = -2; -2 + 5 = 3; 3 - 6 = -3. Yep, that works too.

So, applying the same logic to 100 terms, which is 50 pairs, each giving -1, so total sum is -50.

Alternatively, maybe there's another way to think about it. Let me try to separate the positive and negative terms.

The positive terms are 1, 3, 5, ..., 99. These are the odd numbers from 1 to 99. How many terms are there? Well, since 99 is the 50th odd number (since 1 = 2*1 - 1, 3 = 2*2 - 1, ..., 99 = 2*50 - 1). So there are 50 positive terms.

The negative terms are -2, -4, -6, ..., -100. These are the even numbers from 2 to 100, each multiplied by -1. There are 50 negative terms as well, since 100 is the 50th even number (2*1, 2*2, ..., 2*50).

So the sum is equal to the sum of the first 50 odd numbers minus the sum of the first 50 even numbers.

Let me recall formulas for these sums.

The sum of the first n odd numbers is n². For example, 1 + 3 + 5 + ... + (2n - 1) = n². So here, n = 50, so the sum of positive terms is 50² = 2500.

The sum of the first n even numbers is n(n + 1). For example, 2 + 4 + 6 + ... + 2n = 2(1 + 2 + 3 + ... + n) = 2*(n(n + 1)/2) = n(n + 1). Therefore, the sum of the first 50 even numbers is 50*51 = 2550.

But since the negative terms are -2 -4 -6 ... -100, that's equal to - (sum of first 50 even numbers) = -2550.

Therefore, total sum is sum of positive terms + sum of negative terms = 2500 - 2550 = -50.

Same answer as before. So that confirms it.

Alternatively, maybe another approach using arithmetic series. The original series can be considered as an alternating series. Let me think.

Let me write the series as:

Sum_{k=1}^{100} (-1)^{k+1} * k

Because when k=1, (-1)^{2} *1 = 1, k=2, (-1)^3 *2 = -2, etc.

But maybe I can find a formula for such a sum.

I remember that for an alternating series like this, there's a formula. Let me recall.

Suppose we have Sum_{k=1}^n (-1)^{k+1} *k.

If n is even, then we can pair the terms as (1 - 2) + (3 - 4) + ... + (n-1 - n). Each pair is -1, and there are n/2 pairs. Therefore, total sum is -n/2.

If n is odd, then it's similar but the last term is +n. So the sum would be [ (n-1)/2 * (-1) ] + n.

In our case, n = 100, which is even, so the sum is -100 / 2 = -50.

That's another way to see it.

So regardless of the method I use, grouping terms, separating positive and negative, or using the formula for the alternating series, I get the same answer: -50.

Let me check with another formula. Suppose I use the formula for the sum of an arithmetic series. Wait, but this isn't an arithmetic series because the signs are alternating. But maybe I can express it as two separate series.

Alternatively, think recursively. But that might complicate.

Alternatively, consider that the sum S = 1 - 2 + 3 - 4 + ... + 99 - 100.

If I add S to itself, but shift the terms. Wait, sometimes with alternating series, you can do something like that.

Let me try:

Let me write S = 1 - 2 + 3 - 4 + 5 - ... + 99 - 100

If I add S to itself but shift the second S by one term:

S = 1 - 2 + 3 - 4 + 5 - ... + 99 - 100

S =   1 - 2 + 3 - 4 + 5 - ... + 99 - 100

Adding them:

2S = 1 + (-2 + 1) + (3 - 2) + (-4 + 3) + ... + (99 - 98) + (-100 + 99)

Wait, let me check that. Wait, if I add S + S, but align them:

First S is:

Term 1: 1

Term 2: -2

Term 3: +3

Term 4: -4

...

Term 99: +99

Term 100: -100

If I add S + S, but shift the second S by one term? Wait, maybe another way. Let me think.

Alternatively, for such an alternating series, sometimes multiplying by -1 and adding. But I might be overcomplicating.

Alternatively, since I already have multiple methods giving me -50, perhaps I can just trust that.

But just to try the shifting approach. Let me consider S = 1 - 2 + 3 - 4 + ... + 99 - 100

Let me write S = (1 - 2) + (3 - 4) + ... + (99 - 100) as before, which is 50 terms of -1, total -50.

Alternatively, if I consider S = 1 + (-2 + 3) + (-4 + 5) + ... + (-98 + 99) - 100. Wait, that might not be helpful. Let me check:

Wait, starting from the beginning:

1 + (-2 + 3) + (-4 + 5) + ... + (-98 + 99) - 100.

How many pairs are there? Let's see, starting with 1, then pairs from -2+3, -4+5,... up to -98+99. Then subtract 100.

Each pair (-2+3)=1, (-4+5)=1,..., (-98+99)=1. How many such pairs? From -2 to -98, stepping by -2 each time? Wait, the pairs are (-2 + 3), (-4 + 5), ..., (-98 + 99). Each pair starts at an even number and ends at the next odd number.

The first pair starts at -2, the second at -4,..., the last pair starts at -98. How many terms? The numbers -2, -4,..., -98. That's an arithmetic sequence with first term -2, last term -98, common difference -2.

Number of terms: ((-98) - (-2))/(-2) + 1 = (-96)/(-2) + 1 = 48 + 1 = 49 terms.

Therefore, there are 49 pairs, each contributing +1, so total from the pairs is 49*1 = 49. Then we have the initial 1 and the final -100.

So total sum is 1 + 49 - 100 = 50 - 100 = -50.

Same result.

Therefore, regardless of how I group the terms, I get -50.

Another way: Let me think about the formula for the sum of such an alternating series.

In general, the sum S = 1 - 2 + 3 - 4 + ... + (-1)^{n+1}n.

If n is even, then S = -n/2.

If n is odd, S = (n + 1)/2.

So here, n = 100, even, so S = -100/2 = -50.

Therefore, that's another way to remember.

Alternatively, if I didn't remember that formula, I can derive it.

Let me consider n even. Let n = 2m.

Then S = (1 - 2) + (3 - 4) + ... + (2m - 1 - 2m) = m*(-1) = -m = -n/2.

Exactly. So that works.

So in this case, since n = 100, m = 50, so -50.

Therefore, all methods lead to the same conclusion.

Just to make sure I didn't make a mistake in the formulas. For example, if n is odd, say n = 3: 1 - 2 + 3 = 2. According to the formula, (n + 1)/2 = (3 + 1)/2 = 2. Correct.

Another example: n = 5: 1 - 2 + 3 - 4 + 5 = (1 - 2) + (3 - 4) + 5 = (-1) + (-1) + 5 = 3. Formula: (5 + 1)/2 = 3. Correct.

Therefore, formula holds.

So, confident that for even n, sum is -n/2. Therefore, answer is -50.

Alternatively, if I didn't think of grouping terms, and tried to use the formula for the sum of an arithmetic series. Wait, but the series isn't arithmetic. However, maybe if I consider the positive and negative terms separately.

As I did earlier: sum of positive terms is sum of first 50 odd numbers, which is 50² = 2500. Sum of negative terms is -sum of first 50 even numbers, which is -50*51 = -2550. Then total sum is 2500 - 2550 = -50.

So that's another confirmation.

Alternatively, maybe think of the entire series as the difference between the sum of odd numbers and the sum of even numbers up to 100.

Sum of odd numbers from 1 to 99: as mentioned, 50² = 2500.

Sum of even numbers from 2 to 100: 50*51 = 2550.

Therefore, difference is 2500 - 2550 = -50.

Same result.

Another way: Maybe using the formula for the sum of an arithmetic series.

But since the series alternates, maybe not directly. However, if I separate into two series.

Let me think.

The original series can be considered as the sum of two series: the sum of positive terms and the sum of negative terms.

Positive terms: 1 + 3 + 5 + ... + 99.

Negative terms: -2 -4 -6 - ... -100.

So, sum of positive terms is S1 = 1 + 3 + 5 + ... + 99.

Sum of negative terms is S2 = -2 -4 -6 - ... -100.

Total sum is S1 + S2.

As before.

So S1 is the sum of first 50 odd numbers: 50² = 2500.

S2 is - (2 + 4 + 6 + ... + 100). The sum inside is 2*(1 + 2 + 3 + ... + 50) = 2*(50*51)/2 = 50*51 = 2550.

Therefore, S2 = -2550.

Total sum: 2500 - 2550 = -50.

All right.

Is there another approach? Maybe using recursion or mathematical induction? Probably overkill, but let me try.

Let me suppose that for n even, the sum is -n/2. Let me check for n=2: 1 - 2 = -1, which is -2/2 = -1. Correct.

For n=4: 1 - 2 + 3 - 4 = -2, which is -4/2 = -2. Correct.

Assume that for n=2k, the sum is -k. Then for n=2(k+1), which is 2k+2, the sum would be sum up to 2k + (2k+1) - (2k+2).

Wait, the sum up to 2k is -k, then adding the next two terms: (2k+1) - (2k+2) = -1. Therefore, total sum is -k -1 = -(k + 1) = - ( (2k + 2)/2 ). Which matches the formula.

Therefore, by induction, the formula holds.

Hence, for n=100, which is 2*50, the sum is -50.

So, that's another proof.

Therefore, no matter how I approach this problem, the answer is consistently -50.

Therefore, I can be confident that the answer is -50.

**Final Answer**
The value of the series is \boxed{-50}.

To evaluate the alternating series

$$
1 - 2 + 3 - 4 + 5 - \dots + 99 - 100,
$$

we begin by observing the pattern: the series alternates in sign, starting with a positive and ending with a negative. It continues up to the 100th term, which is $-100$.

---

### Step 1: Group the Terms in Pairs

We can group consecutive pairs of terms:

$$
(1 - 2) + (3 - 4) + (5 - 6) + \cdots + (99 - 100)
$$

Each group simplifies to:

$$
1 - 2 = -1,\quad 3 - 4 = -1,\quad 5 - 6 = -1,\quad \dots,\quad 99 - 100 = -1
$$

So, all the pairs contribute $-1$, and there are 50 such pairs since there are 100 terms in total.

Therefore, the sum becomes:

$$
50 \times (-1) = -50
$$

---

### Step 2: Confirm by Summing Separate Positive and Negative Groups

Alternatively, we can separate the sum into two parts:

- The **positive terms**: $1 + 3 + 5 + \cdots + 99$
- The **negative terms**: $-2 - 4 - 6 - \cdots - 100$

The number of positive terms is the number of odd numbers from 1 to 99, which is 50. The sum of the first $n$ odd numbers is given by the formula:

$$
\text{Sum of first } n \text{ odd numbers} = n^2
$$

So, the sum of the positive terms is:

$$
1 + 3 + 5 + \cdots + 99 = 50^2 = 2500
$$

The negative terms are the first 50 even numbers, from 2 to 100, each with a negative sign. The sum of the first $n$ even numbers is:

$$
\text{Sum of first } n \text{ even numbers} = n(n + 1)
$$

So, the sum of the negative terms is:

$$
-2 - 4 - 6 - \cdots - 100 = - (2 + 4 + 6 + \cdots + 100) = - (50 \times 51) = -2550
$$

Adding these two parts:

$$
2500 - 2550 = -50
$$

---

### Step 3: General Formula for Alternating Series

The general formula for the alternating sum:

$$
\sum_{k=1}^{n} (-1)^{k+1} \cdot k
$$

If $n$ is even, this simplifies to:

$$
\frac{-n}{2}
$$

In our case, $n = 100$, which is even. Applying the formula:

$$
\frac{-100}{2} = -50
$$

---

### Final Answer

All approaches consistently yield the same result. Therefore, the value of the series is:

$$
\boxed{-50}
$$

[Assistant]
Explanation:
Okay, so I need to compute this series: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Let me think about how to approach this.

First, I notice that the series alternates between adding and subtracting numbers. It starts with 1, then subtracts 2, adds 3, subtracts 4, and so on, up to 100. Since the last term is -100, that means there are 100 terms in total. Let me check that. The first term is 1, then each pair of terms is like (1 - 2), (3 - 4), ..., up to (99 - 100). So if I group them in pairs, how many pairs are there?

Well, since there are 100 terms, and each pair has 2 terms, there should be 100 / 2 = 50 pairs. Each pair is of the form (odd number - even number). Let me write out the first few pairs to see the pattern:

(1 - 2) + (3 - 4) + (5 - 6) + ... + (99 - 100)

Each of these pairs simplifies to -1. Let me verify that:

1 - 2 = -1

3 - 4 = -1

5 - 6 = -1

...

99 - 100 = -1

So each pair is indeed -1. Since there are 50 pairs, the total sum would be 50 * (-1) = -50.

Wait, that seems straightforward, but let me make sure I haven't missed anything. Let me check with a smaller example. Let's say I have 1 - 2 + 3 - 4. According to my method, there are 4 terms, so 2 pairs. Each pair is -1, so total is -2. Let me compute manually: 1 - 2 + 3 - 4 = (1 - 2) + (3 - 4) = (-1) + (-1) = -2. That's correct.

Another example: 1 - 2 + 3 - 4 + 5 - 6. That's 3 pairs, so -3. Calculating manually: 1 - 2 = -1; -1 + 3 = 2; 2 - 4 = -2; -2 + 5 = 3; 3 - 6 = -3. Yep, that works too.

So, applying the same logic to 100 terms, which is 50 pairs, each giving -1, so total sum is -50.

Alternatively, maybe there's another way to think about it. Let me try to separate the positive and negative terms.

The positive terms are 1, 3, 5, ..., 99. These are the odd numbers from 1 to 99. How many terms are there? Well, since 99 is the 50th odd number (since 1 = 2*1 - 1, 3 = 2*2 - 1, ..., 99 = 2*50 - 1). So there are 50 positive terms.

The negative terms are -2, -4, -6, ..., -100. These are the even numbers from 2 to 100, each multiplied by -1. There are 50 negative terms as well, since 100 is the 50th even number (2*1, 2*2, ..., 2*50).

So the sum is equal to the sum of the first 50 odd numbers minus the sum of the first 50 even numbers.

Let me recall formulas for these sums.

The sum of the first n odd numbers is n². For example, 1 + 3 + 5 + ... + (2n - 1) = n². So here, n = 50, so the sum of positive terms is 50² = 2500.

The sum of the first n even numbers is n(n + 1). For example, 2 + 4 + 6 + ... + 2n = 2(1 + 2 + 3 + ... + n) = 2*(n(n + 1)/2) = n(n + 1). Therefore, the sum of the first 50 even numbers is 50*51 = 2550.

But since the negative terms are -2 -4 -6 ... -100, that's equal to - (sum of first 50 even numbers) = -2550.

Therefore, total sum is sum of positive terms + sum of negative terms = 2500 - 2550 = -50.

Same answer as before. So that confirms it.

Alternatively, maybe another approach using arithmetic series. The original series can be considered as an alternating series. Let me think.

Let me write the series as:

Sum_{k=1}^{100} (-1)^{k+1} * k

Because when k=1, (-1)^{2} *1 = 1, k=2, (-1)^3 *2 = -2, etc.

But maybe I can find a formula for such a sum.

I remember that for an alternating series like this, there's a formula. Let me recall.

Suppose we have Sum_{k=1}^n (-1)^{k+1} *k.

If n is even, then we can pair the terms as (1 - 2) + (3 - 4) + ... + (n-1 - n). Each pair is -1, and there are n/2 pairs. Therefore, total sum is -n/2.

If n is odd, then it's similar but the last term is +n. So the sum would be [ (n-1)/2 * (-1) ] + n.

In our case, n = 100, which is even, so the sum is -100 / 2 = -50.

That's another way to see it.

So regardless of the method I use, grouping terms, separating positive and negative, or using the formula for the alternating series, I get the same answer: -50.

Let me check with another formula. Suppose I use the formula for the sum of an arithmetic series. Wait, but this isn't an arithmetic series because the signs are alternating. But maybe I can express it as two separate series.

Alternatively, think recursively. But that might complicate.

Alternatively, consider that the sum S = 1 - 2 + 3 - 4 + ... + 99 - 100.

If I add S to itself, but shift the terms. Wait, sometimes with alternating series, you can do something like that.

Let me try:

Let me write S = 1 - 2 + 3 - 4 + 5 - ... + 99 - 100

If I add S to itself but shift the second S by one term:

S = 1 - 2 + 3 - 4 + 5 - ... + 99 - 100

S =   1 - 2 + 3 - 4 + 5 - ... + 99 - 100

Adding them:

2S = 1 + (-2 + 1) + (3 - 2) + (-4 + 3) + ... + (99 - 98) + (-100 + 99)

Wait, let me check that. Wait, if I add S + S, but align them:

First S is:

Term 1: 1

Term 2: -2

Term 3: +3

Term 4: -4

...

Term 99: +99

Term 100: -100

If I add S + S, but shift the second S by one term? Wait, maybe another way. Let me think.

Alternatively, for such an alternating series, sometimes multiplying by -1 and adding. But I might be overcomplicating.

Alternatively, since I already have multiple methods giving me -50, perhaps I can just trust that.

But just to try the shifting approach. Let me consider S = 1 - 2 + 3 - 4 + ... + 99 - 100

Let me write S = (1 - 2) + (3 - 4) + ... + (99 - 100) as before, which is 50 terms of -1, total -50.

Alternatively, if I consider S = 1 + (-2 + 3) + (-4 + 5) + ... + (-98 + 99) - 100. Wait, that might not be helpful. Let me check:

Wait, starting from the beginning:

1 + (-2 + 3) + (-4 + 5) + ... + (-98 + 99) - 100.

How many pairs are there? Let's see, starting with 1, then pairs from -2+3, -4+5,... up to -98+99. Then subtract 100.

Each pair (-2+3)=1, (-4+5)=1,..., (-98+99)=1. How many such pairs? From -2 to -98, stepping by -2 each time? Wait, the pairs are (-2 + 3), (-4 + 5), ..., (-98 + 99). Each pair starts at an even number and ends at the next odd number.

The first pair starts at -2, the second at -4,..., the last pair starts at -98. How many terms? The numbers -2, -4,..., -98. That's an arithmetic sequence with first term -2, last term -98, common difference -2.

Number of terms: ((-98) - (-2))/(-2) + 1 = (-96)/(-2) + 1 = 48 + 1 = 49 terms.

Therefore, there are 49 pairs, each contributing +1, so total from the pairs is 49*1 = 49. Then we have the initial 1 and the final -100.

So total sum is 1 + 49 - 100 = 50 - 100 = -50.

Same result.

Therefore, regardless of how I group the terms, I get -50.

Another way: Let me think about the formula for the sum of such an alternating series.

In general, the sum S = 1 - 2 + 3 - 4 + ... + (-1)^{n+1}n.

If n is even, then S = -n/2.

If n is odd, S = (n + 1)/2.

So here, n = 100, even, so S = -100/2 = -50.

Therefore, that's another way to remember.

Alternatively, if I didn't remember that formula, I can derive it.

Let me consider n even. Let n = 2m.

Then S = (1 - 2) + (3 - 4) + ... + (2m - 1 - 2m) = m*(-1) = -m = -n/2.

Exactly. So that works.

So in this case, since n = 100, m = 50, so -50.

Therefore, all methods lead to the same conclusion.

Just to make sure I didn't make a mistake in the formulas. For example, if n is odd, say n = 3: 1 - 2 + 3 = 2. According to the formula, (n + 1)/2 = (3 + 1)/2 = 2. Correct.

Another example: n = 5: 1 - 2 + 3 - 4 + 5 = (1 - 2) + (3 - 4) + 5 = (-1) + (-1) + 5 = 3. Formula: (5 + 1)/2 = 3. Correct.

Therefore, formula holds.

So, confident that for even n, sum is -n/2. Therefore, answer is -50.

Alternatively, if I didn't think of grouping terms, and tried to use the formula for the sum of an arithmetic series. Wait, but the series isn't arithmetic. However, maybe if I consider the positive and negative terms separately.

As I did earlier: sum of positive terms is sum of first 50 odd numbers, which is 50² = 2500. Sum of negative terms is -sum of first 50 even numbers, which is -50*51 = -2550. Then total sum is 2500 - 2550 = -50.

So that's another confirmation.

Alternatively, maybe think of the entire series as the difference between the sum of odd numbers and the sum of even numbers up to 100.

Sum of odd numbers from 1 to 99: as mentioned, 50² = 2500.

Sum of even numbers from 2 to 100: 50*51 = 2550.

Therefore, difference is 2500 - 2550 = -50.

Same result.

Another way: Maybe using the formula for the sum of an arithmetic series.

But since the series alternates, maybe not directly. However, if I separate into two series.

Let me think.

The original series can be considered as the sum of two series: the sum of positive terms and the sum of negative terms.

Positive terms: 1 + 3 + 5 + ... + 99.

Negative terms: -2 -4 -6 - ... -100.

So, sum of positive terms is S1 = 1 + 3 + 5 + ... + 99.

Sum of negative terms is S2 = -2 -4 -6 - ... -100.

Total sum is S1 + S2.

As before.

So S1 is the sum of first 50 odd numbers: 50² = 2500.

S2 is - (2 + 4 + 6 + ... + 100). The sum inside is 2*(1 + 2 + 3 + ... + 50) = 2*(50*51)/2 = 50*51 = 2550.

Therefore, S2 = -2550.

Total sum: 2500 - 2550 = -50.

All right.

Is there another approach? Maybe using recursion or mathematical induction? Probably overkill, but let me try.

Let me suppose that for n even, the sum is -n/2. Let me check for n=2: 1 - 2 = -1, which is -2/2 = -1. Correct.

For n=4: 1 - 2 + 3 - 4 = -2, which is -4/2 = -2. Correct.

Assume that for n=2k, the sum is -k. Then for n=2(k+1), which is 2k+2, the sum would be sum up to 2k + (2k+1) - (2k+2).

Wait, the sum up to 2k is -k, then adding the next two terms: (2k+1) - (2k+2) = -1. Therefore, total sum is -k -1 = -(k + 1) = - ( (2k + 2)/2 ). Which matches the formula.

Therefore, by induction, the formula holds.

Hence, for n=100, which is 2*50, the sum is -50.

So, that's another proof.

Therefore, no matter how I approach this problem, the answer is consistently -50.

Therefore, I can be confident that the answer is -50.

**Final Answer**
The value of the series is \boxed{-50}.

To evaluate the alternating series

$$
1 - 2 + 3 - 4 + 5 - \dots + 99 - 100,
$$

we begin by observing the pattern: the series alternates in sign, starting with a positive and ending with a negative. It continues up to the 100th term, which is $-100$.

---

### Step 1: Group the Terms in Pairs

We can group consecutive pairs of terms:

$$
(1 - 2) + (3 - 4) + (5 - 6) + \cdots + (99 - 100)
$$

Each group simplifies to:

$$
1 - 2 = -1,\quad 3 - 4 = -1,\quad 5 - 6 = -1,\quad \dots,\quad 99 - 100 = -1
$$

So, all the pairs contribute $-1$, and there are 50 such pairs since there are 100 terms in total.

Therefore, the sum becomes:

$$
50 \times (-1) = -50
$$

---

### Step 2: Confirm by Summing Separate Positive and Negative Groups

Alternatively, we can separate the sum into two parts:

- The **positive terms**: $1 + 3 + 5 + \cdots + 99$
- The **negative terms**: $-2 - 4 - 6 - \cdots - 100$

The number of positive terms is the number of odd numbers from 1 to 99, which is 50. The sum of the first $n$ odd numbers is given by the formula:

$$
\text{Sum of first } n \text{ odd numbers} = n^2
$$

So, the sum of the positive terms is:

$$
1 + 3 + 5 + \cdots + 99 = 50^2 = 2500
$$

The negative terms are the first 50 even numbers, from 2 to 100, each with a negative sign. The sum of the first $n$ even numbers is:

$$
\text{Sum of first } n \text{ even numbers} = n(n + 1)
$$

So, the sum of the negative terms is:

$$
-2 - 4 - 6 - \cdots - 100 = - (2 + 4 + 6 + \cdots + 100) = - (50 \times 51) = -2550
$$

Adding these two parts:

$$
2500 - 2550 = -50
$$

---

### Step 3: General Formula for Alternating Series

The general formula for the alternating sum:

$$
\sum_{k=1}^{n} (-1)^{k+1} \cdot k
$$

If $n$ is even, this simplifies to:

$$
\frac{-n}{2}
$$

In our case, $n = 100$, which is even. Applying the formula:

$$
\frac{-100}{2} = -50
$$

---

### Final Answer

All approaches consistently yield the same result. Therefore, the value of the series is:

$$
\boxed{-50}
$$

Solution: $(1-2)+(3-4)+ \dots +(97-98)+(99-100) = 50(-1) = \boxed{-50}.$