# 2007 UNCO Math Contest II Problems/Problem 7

## Problem

(a) Express the infinite sum $S= 1+ \frac{1}{3}+\frac{1}{3^2}+ \frac{1}{3^3}+ \cdots$ as a reduced fraction.

(b) Express the infinite sum $T=\frac{1}{5}+ \frac{1}{25}+ \frac{2}{125}+ \frac{3}{625}+ \frac{5}{3125}+ \cdots$ as a reduced fraction. Here the denominators are powers of $5$ and the numerators $1, 1, 2, 3, 5, \ldots$ are the Fibonacci numbers $F_n$ where $F_n=F_{n-1}+F_{n-2}$.

## Solution

(a): Knowing that the formula for an infinite geometric series is $A/(1 - r)$, where $A$ and $r$ are the first term and common ratio respectively, we compute $1/(1 - 1/3) = 3/2$, and we have our answer of $3/2$.

(b) $\frac{5}{19}$ $$5T=1+\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{5}{5^4}+\cdots$$ $$T=0+\frac{1}{5}+\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^3}+\cdots$$ $$5T-T=1+0+\frac{1}{5^2}+\frac{1}{5^3}+\frac{2}{5^4}+\cdots = 1+\frac{T}{5}$$

## See Also

 2007 UNCO Math Contest II (Problems • Answer Key • Resources) Preceded byProblem 6 Followed byProblem 8 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 All UNCO Math Contest Problems and Solutions
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