The formula
The sum of 1 through n is n x (n + 1) / 2. Example: 1 + 2 + ... + 50 = 50 x 51 / 2 = 1275.
Why it works
Pair the first and last numbers, then the second and second-to-last, and so on. Each pair adds to n + 1, and there are n / 2 pairs. For 1 through 100, every pair adds to 101 and there are 50 pairs, giving 5050.
Sums of odd and even numbers
The first n odd numbers add to n^2. The first n even numbers add to n x (n + 1). Example: 2 + 4 + ... + 40 has 20 even numbers, so the sum is 20 x 21 = 420.
Sums that do not start at 1
Use first plus last, times the count, divided by 2. Example: 11 + 12 + ... + 20 has 10 numbers, so the sum is (11 + 20) x 10 / 2 = 155.
Practice problems
- 1 + 2 + ... + 20
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20 x 21 / 2 = 210.
- 1 + 2 + ... + 100
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100 x 101 / 2 = 5050.
- 1 + 3 + 5 + ... + 19
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There are 10 odd numbers, so the sum is 10^2 = 100.
References
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