Add all the powers of two up to 512

Question

Solve in your head. What is 1 + 2 + 4 + 8 + \dots + 512? Aim: under 10 seconds.

Accepted Answer

This is the doubling shortcut with first term 1: 	ext{sum} = 2 	imes 	ext{last} - 	ext{first} = 2 	imes 512 - 1 = 1024 - 1 = 1023. The cleaner way to see it: 1 + 2 + 4 + \dots + 2^k = 2^{k+1} - 1. Here 512 = 2^9, so the sum is 2^{10} - 1 = 1024 - 1 = 1023. In binary, adding all the powers of two up to 2^9 turns on every bit from position 0 to 9, which is 1023 = \underbrace{1111111111}{10 	ext{ ones}}, exactly one below the next power of two. Takeaway. The sum of powers of two from 1 to 2^k is 2^{k+1} - 1: a row of ones in binary, always one short of the next power. That "all ones is one…

Add all the powers of two up to 512

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