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Algebra Level 4

∑ n = 0 ∞ 3 n + 5 n 8 n = a b \large \sum_{n = 0}^\infty \frac{3^n + 5^n}{8^n} = \frac ab

The equation above holds true for some coprime positive integers a a and b b . Find a + b a+b .


The answer is 79.

Math Man by math man , 20, Indonesia

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1 solution

∑ n = 0 ∞ 3 n + 5 n 8 n = ∑ n = 0 ∞ ( 3 8 ) n + ∑ n = 0 ∞ ( 5 8 ) n = 1 1 − 3 8 + 1 1 − 5 8 = 8 5 + 8 3 = 64 15 \begin{aligned} \sum_{n=0}^\infty \frac {3^n+5^n}{8^n} & = \sum_{n=0}^\infty \left(\frac 38\right)^n + \sum_{n=0}^\infty \left(\frac 58\right)^n \\ & = \frac 1{1-\frac 38} + \frac 1{1-\frac 58} \\ & = \frac 85 + \frac 83 \\ & = \frac {64}{15} \end{aligned}

⟹ a + b = 64 + 15 = 79 \implies a+b = 64+15 = \boxed{79} .

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