Question

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

for all positive values of $a_1$ and $a_2$

if and only if $a_1=a_2$

if and only if $a_1=1$

if and only if $a_2=1$

if and only if $a_1=a_2=1$

Answer: E

Concepts:recursion geometric sequence ratio and proportion

Difficulty rating: 1860

Small Hint:

Write out $a_3,$ $a_4,$ and $a_5$ in terms of $a_1,$ $a_2$

Big Hint:

Equate the first three successive ratios and use positivity

Solution:

The next terms are $a_3=a_1a_2,$ $a_4=a_1a_2^2,$ and $a_5=a_1^2a_2^3.$ If the sequence is geometric, then $\begin{aligned} \frac{a_2}{a_1}&=\frac{a_3}{a_2}=a_1,\\ \frac{a_4}{a_3}&=a_2=a_1. \end{aligned}$ Thus $a_2=a_1^2$ and $a_2=a_1.$ Positivity forces $a_1=a_2=1.$ Conversely, these initial values produce the constant geometric sequence $1,1,1,\ldots.$

Therefore, the correct answer is E.

1977 AMC 12 Problem 13

Answer: E

Difficulty rating: 1860

Small Hint:

Big Hint:

Solution:

Problem 13 in Other Years