1960 AMC 12 Problem 36

Question

Let

s_1,

s_2,

s_3

be the respective sums of

n,

2n,

3n

terms of the same arithmetic progression with

a

as the first term and

d

as the common difference. Let

R=s_3-s_2-s_1.

Then

R

is dependent on:

a

and

d

d

and

n

a

and

n

a,

d,

and

n

neither

a

nor

d

nor

n

Use

S_k=\dfrac{k}{2}(2a+(k-1)d)

Substitute

k=n,2n,3n

and collect the

a

-terms and

d

-terms separately

Using the arithmetic-series formula,

s_j=\frac{jn}{2}\bigl(2a+(jn-1)d\bigr).

In

s_3-s_2-s_1,

the coefficient of

a

is

3n-2n-n=0.

Simplifying the remaining terms gives

R=2n^2d.

Thus

R

depends on

d

and

n,

but not on

a.

Therefore, the correct answer is B.

Problem 36 in other years