Difference between revisions of "1951 AHSME Problems/Problem 11"
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Let the original geometric series be <math>a,ar,ar^2,ar^3,ar^4\cdots</math>. Therefore, their squares are <math>a^2,a^2r^2,a^2r^4,a^2r^6,\cdots</math>, which is a [[Geometric sequence|geometric sequence]] with first term <math>a^2</math> and common ratio <math>r^2</math>. Thus, the sum is <math>\boxed{\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2}}</math>. | Let the original geometric series be <math>a,ar,ar^2,ar^3,ar^4\cdots</math>. Therefore, their squares are <math>a^2,a^2r^2,a^2r^4,a^2r^6,\cdots</math>, which is a [[Geometric sequence|geometric sequence]] with first term <math>a^2</math> and common ratio <math>r^2</math>. Thus, the sum is <math>\boxed{\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2}}</math>. | ||
− | == See | + | == See Also == |
− | {{AHSME box|year=1951|num-b=10|num-a=12}} | + | {{AHSME 50p box|year=1951|num-b=10|num-a=12}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 08:51, 29 April 2012
Problem
The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 \minus{} r}$ (Error compiling LaTeX. ! Undefined control sequence.) where denotes the first term and $\minus{} 1 < r < 1$ (Error compiling LaTeX. ! Undefined control sequence.) denotes the common ratio. The limit of the sum of their squares is:
$\textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$ (Error compiling LaTeX. ! Undefined control sequence.)
Solution
Let the original geometric series be . Therefore, their squares are , which is a geometric sequence with first term and common ratio . Thus, the sum is $\boxed{\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2}}$ (Error compiling LaTeX. ! Undefined control sequence.).
See Also
1951 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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All AHSME Problems and Solutions |