1956 AMC 12 Problem 25

Attempt Problem 25 of the 1956 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 1956 AMC 12 solutions, or check the answer key.

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25.

The sum of all numbers of the form

2k+1,

where

k

takes on integral values from

1

n

is:

n^2

n(n+1)

n(n+2)

(n+1)^2

(n+1)(n+2)

Separate

\sum_{k=1}^n(2k+1)

into two sums

Use

1+2+\cdots+n=\frac{n(n+1)}{2}

We compute

\begin{aligned} \sum_{k=1}^n(2k+1) &=2\sum_{k=1}^n k+\sum_{k=1}^n1\\ &=n(n+1)+n\\ &=n(n+2). \end{aligned}

Thus, the correct answer is C.

Problem 25 in other years