Sum a tripling series 2 + 6 + 18 + … + 486

Question

Solve in your head. What is 2 + 6 + 18 + 54 + 162 + 486? Aim: under 15 seconds.

Accepted Answer

Each term is 3 times the previous, so the ratio is r = 3. The general shortcut for a finite geometric series is 	ext{sum} = \frac{r 	imes 	ext{last} - 	ext{first}}{r - 1}. Plug in last = 486, first = 2, r = 3: 	ext{sum} = \frac{3 	imes 486 - 2}{3 - 1} = \frac{1458 - 2}{2} = \frac{1456}{2} = 728. Adding the terms directly confirms it: 2 + 6 + 18 + 54 + 162 + 486 = 728. Notice the doubling rule is just this formula at r = 2, where the denominator r - 1 becomes 1 and it collapses to 2 	imes 	ext{last} - 	ext{first}. Takeaway. For any ratio r, a finite geometric series sums to \dfrac{r\e…

Sum a tripling series 2 + 6 + 18 + … + 486

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