1959 AMC 12 Problem 37

Attempt Problem 37 of the 1959 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 1959 AMC 12 solutions, or check the answer key.

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37.

When simplified the product

\begin{aligned} &\left(1-\dfrac13\right)\left(1-\dfrac14\right)\\ &\quad{}\cdot\left(1-\dfrac15\right)\cdots\left(1-\dfrac1n\right) \end{aligned}

becomes:

\dfrac1n

\dfrac2n

\dfrac{2(n-1)}n

\dfrac{2}{n(n+1)}

\dfrac{3}{n(n+1)}

Answer: B

Concepts:telescoping fraction

Difficulty rating: 1140

Small hint:

Rewrite each factor

1-\frac{1}{k}

as

\frac{k-1}{k}

Big hint:

Write out the first few factors and cancel adjacent numerators and denominators

Solution:

The product telescopes:

\frac23\cdot\frac34\cdot\frac45\cdots\frac{n-1}{n} =\frac2n.

Therefore, the correct answer is B.

Problem 37 in other years