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Difference between revisions of "1977 AHSME Problems/Problem 13"

(Created page with "== Problem 13 == If <math>a_1,a_2,a_3,\dots</math> is a sequence of positive numbers such that <math>a_{n+2}=a_na_{n+1}</math> for all positive integers <math>n</math>, then...")
 
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Solution by e_power_pi_times_i
 
Solution by e_power_pi_times_i
  
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The first few terms are <math>a_1,a_2,a_1a_2,a_1a_2^2,a_1^2a_2^3,\dots</math>. If this is a geometric progression, <math>\dfrac{a_2}{a_1} = a_1 = a_2 = a_1a_2</math>. <math>a_1=0,1</math>, <math>a_2=0,1</math>. Since this is a sequence of positive integers, then the answer must be <math>\boxed{\textbf{(E) }\text{if and only if }a_1=a_2=1 }</math>
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The first few terms are <math>a_1,a_2,a_1a_2,a_1a_2^2,a_1^2a_2^3,\dots</math>. If this is a geometric progression, <math>\dfrac{a_2}{a_1} = a_1 = a_2 = a_1a_2</math>. <math>a_1=0,1</math>, <math>a_2=0,1</math>. Since this is a sequence of positive integers, then the answer must be <math>\boxed{\textbf{(E) }\text{if and only if }a_1=a_2=1 }</math>.

Revision as of 12:15, 21 November 2016

Problem 13

If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression

$\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad\\ \textbf{(B) }\text{if and only if }a_1=a_2\qquad\\ \textbf{(C) }\text{if and only if }a_1=1\qquad\\ \textbf{(D) }\text{if and only if }a_2=1\qquad\\ \textbf{(E) }\text{if and only if }a_1=a_2=1$


Solution

Solution by e_power_pi_times_i

The first few terms are $a_1,a_2,a_1a_2,a_1a_2^2,a_1^2a_2^3,\dots$. If this is a geometric progression, $\dfrac{a_2}{a_1} = a_1 = a_2 = a_1a_2$. $a_1=0,1$, $a_2=0,1$. Since this is a sequence of positive integers, then the answer must be $\boxed{\textbf{(E) }\text{if and only if }a_1=a_2=1 }$.

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