Add up every date in a 31-day month

Question

Take a calendar page for a 31-day month and add up all the date numbers, 1 + 2 + 3 + ... + 31.

What is the total, and what is the fastest way to get it without adding one by one? As a bonus, note anything special about the number you get.

Accepted Answer

496. Pairing. 1 + 31 = 32, 2 + 30 = 32, ..., 15 + 17 = 32: that is 15 pairs of 32, which is 480, plus the unpaired middle number 16. Total 480 + 16 = 496. Formula. The sum of 1 to n is n(n + 1)/2. For n = 31: 31 × 32 / 2 = 31 × 16 = 496. For other month lengths. 30 days: 30 × 31 / 2 = 465. 29 days: 435. 28 days: 406. The bonus. 496 is a perfect number: it equals the sum of its proper divisors, 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496. The first few perfect numbers are 6, 28 and 496, and it is no coincidence that 28 (the days in February) and 496 (the sum of dates in a long month) both s…

Add up every date in a 31-day month

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