1958 AMC 12 Problem 37

Question

The first term of an arithmetic series of consecutive integers is

k^2+1.

The sum of

2k+1

terms of this series may be expressed as:

k^3+(k+1)^3

(k-1)^3+k^3

(k+1)^3

(k+1)^2

(2k+1)(k+1)^2

Find the last term after

2k

increases from the first term

Use the arithmetic-series average of the first and last terms

The last term is

k^2+1+2k=(k+1)^2.

The average of the first and last terms is

k^2+k+1.

Therefore the sum is

\begin{aligned} S&=(2k+1)(k^2+k+1)\\ &=k^3+(k+1)^3. \end{aligned}

Therefore, the correct answer is A.

Problem 37 in other years