1977 AHSME Problems/Problem 13

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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1977 AHSME Problems/Problem 13

Problem 13

Solution