1953 AMC 12 Problem 25

Attempt Problem 25 of the 1953 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 1953 AMC 12 solutions, or check the answer key.

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25.

In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. Then the common ratio is:

1

about

\dfrac{\sqrt5}{2}

\dfrac{\sqrt5-1}{2}

\dfrac{1-\sqrt5}{2}

\dfrac2{\sqrt5}

Divide the relation among three consecutive terms by the first of them

The common ratio satisfies

1=r+r^2

; choose its positive root

If a term is

t,

the next two are

tr

and

tr^2.

Thus

t=tr+tr^2

and

r^2+r-1=0.

Because the terms are positive,

r=\frac{\sqrt5-1}{2}.

Thus, the correct answer is C.

Problem 25 in other years