1962 AMC 12 Problem 32

Attempt Problem 32 of the 1962 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 1962 AMC 12 solutions, or check the answer key.

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32.

If

x_{k+1}=x_k+\dfrac12

for

k=1,

2,

\ldots,

n-1

and

x_1=1,

find

x_1+x_2+\cdots+x_n.

\dfrac{n+1}{2}

\dfrac{n+3}{2}

\dfrac{n^2-1}{2}

\dfrac{n^2+n}{4}

\dfrac{n^2+3n}{4}

Answer: E

Concepts:arithmetic sequence summation

Difficulty rating: 1440

Small hint:

The recurrence defines an arithmetic sequence with common difference

\frac{1}{2}

Big hint:

Find

x_n

, then use

\frac{n(x_1+x_n)}{2}

Solution:

We have

x_n=1+\frac{n-1}{2}=\frac{n+1}{2}.

Therefore

\begin{aligned} x_1+\cdots+x_n &=\frac n2\left(1+\frac{n+1}{2}\right)\\ &=\frac{n^2+3n}{4}. \end{aligned}

Thus, the correct answer is E.