1977 AMC 12 Problem 13

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

Write out

a_3,

a_4,

and

a_5

in terms of

a_1,

a_2

Equate the first three successive ratios and use positivity

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.

Problem 13 in other years