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Question: What is the sum of all the integers, inclusive, from 1 to 100? A) 942 B) 1010 C) 3031 D) 5050 E) 10,100

## What’s the solution to this SAT Math question?

We are asked to add (“the sum”) of all integers (whole numbers and 0) from 1 to 100, including 1 and 100 (“inclusive” means counting the first and last terms, and “exclusive” would be didn’t count 1 and 100).

Ok, sub-quiz: What’s the first thing you do to solve that math problem? 1) Scribble a 100-row addition problem down by hand 2) Furiously tap away at your TI-83 or variation thereof – it’s only a little more than 300 keystrokes! 3) Calmly take out your pencil, write down “1+2+3…+98+99+100” and consider what you see for a moment before solving the problem with a shortcut.

Yup! Choice 3 is the right answer! Like many sequence problems, there’s a huge shortcut on this question.

## What’s the secret SAT math shortcut?

Our list of numbers to add is “1+2+3…+98+99+100”

Look at the outermost numbers of the sequence, 1 and 100. We’re being asked to “sum” everything, right? What’s 1+100? 101, right? Now pair up the next-most outer numbers, 2 and 99. What’s 2+99? 101? Wait a second… for the next-most outer pair of numbers, we have 3 and 98. What’s 3+98? 101 again!

That’s right, we officially have a clear pattern here (it’s also important that you have the skills and experience to make the connection between the words and the math!). There’s a repeated unit of 2 numbers that always adds up to “101.” We just gotta know how many “101”s we have.

You may already have jumped to the conclusion that we have 50 pairs of 2 numbers each, for a total of the 1 to 100 we started with. And 50 pairs x 101 per pair is 5050 total.