1971 AMC 12 Problem 33

Attempt Problem 33 of the 1971 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 1971 AMC 12 solutions, or check the answer key.

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33.

If

P

is the product of

n

quantities in geometric progression,

S

their sum, and

S'

the sum of their reciprocals, then

P

in terms of

S,

S',

and

n

is:

(SS')^{\frac{n}{2}}

(\frac{S}{S'})^{\frac{n}{2}}

(SS')^{n-2}

(\frac{S}{S'})^n

(\frac{S}{S'})^{\frac{n-1}{2}}

Answer: B

Concepts:geometric sequence algebraic manipulation

Difficulty rating: 2300

Small hint:

Write the progression as

a,ar,\ldots,ar^{n-1}

Big hint:

Show that

\frac{S}{S'}=a^2r^{n-1}

, whose

\frac{n}{2}

power is the product

Solution:

Write the terms as

a,ar,\ldots,ar^{n-1}.

Reversing the reciprocal sum gives

S'=\frac{S}{a^2r^{n-1}},

so

\frac{S}{S'}=a^2r^{n-1}.

Meanwhile

\begin{aligned} P&=a^nr^{\frac{n(n-1)}{2}}\\ &=\left(a^2r^{n-1}\right)^{\frac{n}{2}}\\ &=\left(\frac{S}{S'}\right)^{\frac{n}{2}}. \end{aligned}

Therefore, the correct answer is B.