Carson,
This is a great question, and YOU WILL HAVE ONE ON YOUR TEST. I've had one on every test I've taken. I've written out a lot, but read it all and you'll see how simple it is at the end.
it's all the same formula. Remember this: the average of consecutive numbers is always the number in the middle.
For example: 2,3,4,5,6
the average is 4. That means since there are 5 numbers, the sum must be 4x5=20.
Or this: 2,3,4,5
The average is 3.5, and the sum is 3.5x4=14
If you've got even numbers or odd numbers, it's the same thing:
2,4,6,8,10; avg=6, total=6x5=30
3,5,7,9,11; avg=7, total=7x5=35
That's the first part of your question. If you have long lists, we have to take another step, because finding the middle isn't always easy. Look at this:
1,2,3,4,5,6,7,8,9
The average is 5, since it's in the middle, so the sum would be 5x9=45.
But notice that the average of JUST THE ENDPOINTS, 1 and 9, is also 5. So you could say, what is the sum of all the numbers 1-9, and without writing them all down, just average 1 and 9, and you'll get the middle.
Now we need to know how many numbers there are, and if you've studied the word "inclusive," you know to subtract and add 1.
So if I asked what the sum of all the numbers from 23-75 are, we'd start with (23+75)/2 = 49 (that's the average of them all), and the number of numbers would be 75-23+1=53. So the sum is 49x53=2597.
The last thing to note is that if you have consecutive even numbers, or consecutive odds, after you subtract, you have to divide by 2, and then add one. There are 7 even integers from 22-34, because 34-12=12/2=6+1=7.
So finally, the answers to the questions you posed:
13-77:
Step 1: (77+13)/2=45
Step 2: 77-13=64/2=32+1=33
Step 3: 45x33=1485
12-88:
Step 1: (12+88)/2=50
Step 2: 88-12=76/2=38+1=39
Step 3: 50x39=1950
I know you can't get it for some reason, but I have this all laid out on my website if you want to see it live:
http://www.integratedlearning.net/city_la_ian_video55.shtml