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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.
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The answer is D, 5050 That’s how you add 100 numbers in 10 seconds! (There’s an old math history urban legend behind this story… who wants to dig it up first?)
SAT Math Mastery Level 1: Perfect Score Fundamentals
SAT Math Mastery Level 2: Tougher Tricks and Skills
Power 800 SAT Math Video Course
The Official SAT Study Guide, ed. 2016
PWN the SAT Math Guide (Review Article)
“Translating Words Into Math” (Blog Article)
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