Infinite Geometric Series

Algebra Level 3

∑ n = 1 ∞ ( 5 8 ) n − 2 = ? \sum _{ n=1 }^{ \infty }{ \left (\frac { 5 }{ 8 } \right)^{ n-2 } } = \, ?

512/75 64/15 5/3 65/64

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Liam P by Liam P , 22, Canada

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2 solutions

Rahil Sehgal
Mar 19, 2017

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Akeel Howell
Mar 18, 2017

Let ∑ n = 1 ∞ ( 5 8 ) n − 2 = a \displaystyle\sum_{n=1}^{\infty}{\left(\dfrac{5}{8}\right)}^{n-2} =\ a

⟹ 8 5 a = 64 25 + a so 3 5 a = 64 25 ⟹ a = 64 15 \implies \dfrac{8}{5}a = \dfrac{64}{25}+a \\ \text{so }\quad \dfrac{3}{5}a = \dfrac{64}{25} \implies a =\ \boxed{\dfrac{64}{15}}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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