The increasing sequence of positive integers a_1, a_2, a_3, … has the property that, for every n≥1, a_n+2=a_n+a_n+1. If a_7=120, then a_8 is

1992 AMC 12 Problem 18

Attempt Problem 18 of the 1992 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1992 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

The increasing sequence of positive integers

a_1,

a_2,

a_3,

\ldots

has the property that, for every

n\ge1,

a_{n+2}=a_n+a_{n+1}.

a_7=120,

then

a_8

128

168

193

194

210

Write

a_7

and

a_8

in terms of

a_1

and

a_2

From

5a_1+8a_2=120,

use congruences and

a_2\gt a_1\gt0

Let

a_1=a

and

a_2=b.

Repeated use of the recurrence gives

\begin{aligned} a_7&=5a+8b=120,\\ a_8&=8a+13b. \end{aligned}

Modulo

5,

the first equation shows that

b

is a multiple of

5.

The positive possibilities are

b=5,

b=10,

b=15.

The conditions

b\gt a\gt0

leave only

b=10

and

a=8.

Hence

a_8=8(8)+13(10)=194.

Thus the correct answer is D.