1957 AMC 12 Problem 30

Question

The sum of the squares of the first

n

positive integers is given by the expression

\dfrac{n(n+c)(2n+k)}6,

if

c

and

k

are, respectively:

1

and

2

3

and

5

2

and

2

1

and

1

2

and

1

Compare the expression with the standard sum-of-squares formula

Alternatively, substitute

n=1

and

n=2

to obtain two equations

The standard formula is

\begin{aligned} 1^2+2^2+\cdots+n^2 &=\frac{n(n+1)}6\\ &\quad\cdot(2n+1). \end{aligned}

Therefore

c=1

and

k=1.

Thus, the correct answer is D.

Problem 30 in other years