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 $n$

Answer: B

Concepts:arithmetic sequence summation algebraic manipulation

Difficulty rating: 1670

Small Hint:

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

Big Hint:

Substitute $k=n,2n,3n$ and collect the $a$ -terms and $d$ -terms separately

Solution:

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.

1960 AMC 12 Problem 36

Answer: B

Difficulty rating: 1670

Small Hint:

Big Hint:

Solution:

Problem 36 in Other Years