2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 15 Medium

Let S_{n} be the sum of the first n term 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}?

  • A.

    340

  • B.

    360

  • C.

    380

  • D.

    400

  • E.

    420

Answer:D

\begin{aligned} \frac{S_{3 n}}{S_n} & =\frac{[a+a+2 \times(3 n-1)] \times 3 n}{[a+a+2 \times(n-1)] \times n} \\ & =\frac{3[2 a+2(3 n-1)]}{2 a+2(n-1)} \\ & =\frac{18 n+6 a-6}{2 n+2 a-2}\end{aligned}.

Since the quotient \dfrac{S_{3n}}{S_{n}} does not depend on n, a=1, and S_{20}=1+3+5+\cdots+39=400.

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