Answer:
200
Step-by-step explanation:
We have: 

We can rearrange the numbers to obtain: 

From the left, we can factor out a negative. So: 

In other words, we want to find the sum of all the odd numbers from 1 to 99. 
And the sum of all the even numbers from 2 to 100. 
Let's do each one individually: 
Odd Terms: 
We have: 

We can use the arithmetic series formula, where: 

Where k is the number of terms, a is the first term, and x_k is the last term. 
Since it's all the odd numbers between 1 and 99, there are 50 terms. 
Our first term is 1 and our last term is 99. So, the sum of all the odd terms are: 

Divide the fraction. Add within the parentheses: 

Multiply: 

So, the sum of all the odd terms is 2500. 
Even Terms: 
We have: 

Again, we can use the above formula. 
Our first term is 2, last term is 100. And since it's from 2-100, we have 50 even terms. So: 

Divide and add: 

Multiply: 

We originally had: 

Substitute them for their respective sums: 

Multiply: 

Add: 

Multiply: 

So, the sum of our sequence is 200. 
And we're done!
Note: I just found a <em>way</em> easier way to do this. We have: 

Let's group every two terms together. So: 

We can see that they each sum to 1: 

Since there are 100 terms, we will have 50 pairs, so 50 times 1. So: 

Multiply: 

Pick which one you want to use! I will suggest this one though...
Edit: Typo