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GMAT Quant
Common Sums and Consecutive Integers
Use compact formulas for consecutive sums and exploit the average of consecutive integers.
Core sum
The sum from 1 through n has a fixed formula.
1 + 2 + ... + n = n(n+1)/2.
Example: 1 through 40
- n
- 40
- Formula
- 40*41/2
- Simplify
- 20*41
- Sum
- 820
Partial range
To sum from a through b, subtract the unwanted beginning.
Sum a..b = sum 1..b - sum 1..(a-1).
Example: 12 through 30
- Sum 1..30
- 30*31/2 = 465
- Sum 1..11
- 11*12/2 = 66
- Sum 12..30
- 399
Special sum
The first n odd numbers sum to n^2.
This pattern makes some odd-number sums immediate.
Special sum
The first n even numbers sum to n(n+1).
For 2 + 4 + ... + 2n, factor out 2 and use the 1-through-n formula.
Consecutive integers
Consecutive integers are centered on their average.
For an odd number of consecutive integers, the average is the middle integer. For an even count, the average lies halfway between the two middle integers.
Mini application
Try this: five consecutive integers sum to 115. What are they?
Find the average first.
Solution
They are 21, 22, 23, 24, 25.
The average is 115/5 = 23, so place two integers on each side.
Remember this
Use symmetry around the average.
For consecutive or evenly spaced numbers, the center often reveals the whole set faster than building equations.