Difference between revisions of "2007 UNCO Math Contest II Problems/Problem 7"
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== Solution == | == Solution == | ||
− | + | (a): Knowing that the formula for an infinite geometric series is <math>A/(1 - r)</math>, where <math>A</math> and <math>r</math> are the first term and common ratio respectively, we compute <math>1/(1 - 1/3) = 3/2</math>, and we have our answer of <math>3/2</math>. | |
+ | (b) <math>\frac{5}{19}</math> | ||
+ | <cmath>5T=1+\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{5}{5^4}+\cdots</cmath> | ||
+ | <cmath>T=0+\frac{1}{5}+\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^3}+\cdots</cmath> | ||
− | {{ | + | <cmath>5T-T=1+0+\frac{1}{5^2}+\frac{1}{5^3}+\frac{2}{5^4}+\cdots = 1+\frac{T}{5}</cmath> |
== See Also == | == See Also == |
Latest revision as of 05:26, 12 January 2019
Problem
(a) Express the infinite sum as a reduced fraction.
(b) Express the infinite sum as a reduced fraction. Here the denominators are powers of and the numerators are the Fibonacci numbers where .
Solution
(a): Knowing that the formula for an infinite geometric series is , where and are the first term and common ratio respectively, we compute , and we have our answer of .
(b)
See Also
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 |