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jmc

algebra senior

Problem

Let

F_{n}

be the Fibonacci sequence, that is,

F_{0} = 0

,

F_{1} = 1

, and

F_{n + 2} = F_{n + 1} + F_{n}

. Compute

n = 0 \sum \infty \frac{F _{n}}{1 0 ^{n}} .

Solution

Let

S = \sum_{n = 0}^{\infty} \frac{F _{n}}{1 0 ^{n}} .

Then

S = F_{0} + \frac{F _{1}}{10} + \frac{F _{2}}{1 0 ^{2}} + \frac{F _{3}}{1 0 ^{3}} + \dots = \frac{F _{0} + 1}{10} + \frac{F _{1} + F _{0}}{1 0 ^{2}} + \frac{F _{2} + F _{1}}{1 0 ^{3}} + \dots = \frac{1}{10} + \frac{F _{0}}{10} + \frac{F _{1}}{1 0 ^{2}} + \frac{F _{2}}{1 0 ^{3}} + \dots + \frac{F _{0}}{1 0 ^{2}} + \frac{F _{1}}{1 0 ^{3}} + \dots = \frac{1}{10} + \frac{1}{10} S + \frac{1}{1 0 ^{2}} S .

Solving, we find

S = \frac{10}{89} .

Final answer

\frac{10}{89}