2022 AMC 10B Problem 15

Question

Let

S_n

be the sum of the first

n

terms of an arithmetic sequence that has a common difference of

2.

The quotient

\dfrac{S_{3n}}{S_n}

does not depend on

n.

What is

S_{20}?

340

360

380

400

420

Force the quotient to be independent of n

Write the arithmetic sequence in terms of first term and common difference

Write the

n

th term as

a+2n,

so the term before the first term is

a.

Then

\begin{aligned}S_n&=\sum_{i=1}^n(a+2i)\\&=n(a+n+1).\end{aligned}

Hence

\begin{aligned} \frac{S_{3n}}{S_n}&=\frac{3(a+3n+1)}{a+n+1}\\ &=9-\frac{6(a+1)}{a+n+1}. \end{aligned}

For this expression to be independent of

n,

its numerator

6(a+1)

in the final fraction must be

0.

Thus

a=-1.

Therefore,

S_{20}=20(-1+20+1)=400.

Thus, the answer is D .

Problem 15 in other years