Sum 5 + 10 + 20 + … + 640 fast

Question

Solve in your head. What is 5 + 10 + 20 + 40 + \dots + 640? Aim: under 10 seconds.

Accepted Answer

For a ratio-2 geometric series, the sum is one doubling past the last term, minus the first term: 	ext{sum} = 2 	imes 	ext{last} - 	ext{first} = 2 	imes 640 - 5 = 1280 - 5 = 1275. Why: each partial sum of doublings equals "the next term minus the first," just as 1 + 2 + 4 + \dots + 2^k = 2^{k+1} - 1. You never have to count how many terms there are; the last term carries all the information. As a check, the terms are 5, 10, 20, 40, 80, 160, 320, 640, and adding them gives 1275. Takeaway. Doubling series: sum = 2\ell - a, where \ell is the last term and a the first. The starting value can…

Sum 5 + 10 + 20 + … + 640 fast

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