2007 UNCO Math Contest II Problems/Problem 7

Revision as of 08:50, 30 January 2018 by Mitko pitko (talk | contribs) (Solution)

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 we have our answer of $3/2$.


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See Also

2007 UNCO Math Contest II (ProblemsAnswer KeyResources)
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
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