Sum 3 + 6 + 12 + … + 384 at a glance

Question

Solve in your head. What is 3 + 6 + 12 + 24 + \dots + 384? Aim: under 10 seconds.

Accepted Answer

For a ratio-2 geometric series, the sum is one step past the last term, minus the first term: 	ext{sum} = 2 	imes 	ext{last} - 	ext{first} = 2 	imes 384 - 3 = 768 - 3 = 765. Why: each partial sum of doublings is "the next term minus the first term", just as 1+2+4+\dots+2^k = 2^{k+1}-1. Takeaway. Doubling series: sum = 2\ell - a where \ell is the last term and a the first. No need to count terms or invoke a\frac{r^n-1}{r-1}. The cousin worth knowing: for ratio 	frac12 summed forever, sum = 2a, which is why 1 + 	frac12 + 	frac14 + \dots = 2 and 	frac14 + 	frac1{16} + 	frac1{64} + \do…

Sum 3 + 6 + 12 + … + 384 at a glance

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