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

Part 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 therefore, we have our answer of $2/3$ .

This problem needs a solution. If you have a solution for it, please help us out by adding it.

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

Page

Toolbox

Search

2007 UNCO Math Contest II Problems/Problem 7

Problem

Solution

See Also