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Difference between revisions of "2016 AMC 12B Problems/Problem 14"

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==Solution==
 
==Solution==
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The second term in a geometric series is <math>a_2 = a \cdot r</math>, where <math>r</math> is the common ratio for the series and <math>a</math> is the first term of the series. So we know that <math>a\cdot r = 1</math> and we wish to find the minimum value of the infinite sum of  the series. We know that: <math>S_\infty = \frac{a}{1-r}</math> and substituting in <math>a=\frac{1}{r}</math>, we get that <math>S_\infty = \frac{\frac{1}{r}}{1-r} = \frac{1}{1-r^2}</math>.
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The second term in a geometric series is <math>a_2 = a \cdot r</math>, where <math>r</math> is the common ratio for the series and <math>a</math> is the first term of the series. So we know that <math>a\cdot r = 1</math> and we wish to find the minimum value of the infinite sum of  the series. We know that: <math>S_\infty = \frac{a}{1-r}</math> and substituting in <math>a=\frac{1}{r}</math>, we get that <math>S_\infty = \frac{\frac{1}{r}}{1-r} = \frac{1}{r(1-r)</math> = \frac{1}{r}+\frac{1-r}$.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=B|num-b=13|num-a=15}}
 
{{AMC12 box|year=2016|ab=B|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 11:16, 21 February 2016

Problem

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$

$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

The second term in a geometric series is $a_2 = a \cdot r$, where $r$ is the common ratio for the series and $a$ is the first term of the series. So we know that $a\cdot r = 1$ and we wish to find the minimum value of the infinite sum of the series. We know that: $S_\infty = \frac{a}{1-r}$ and substituting in $a=\frac{1}{r}$, we get that $S_\infty = \frac{\frac{1}{r}}{1-r} = \frac{1}{r(1-r)$ (Error compiling LaTeX. Unknown error_msg) = \frac{1}{r}+\frac{1-r}$.

See Also

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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