Let s_1 be the sum of the first n terms of the arithmetic sequence 8, 12, … and let s_2 be the sum of the first n terms of the arithmetic sequence 17, 19, …. Then s_1=s_2 for:

1966 AMC 12 Problem 19

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19.

Let

s_1

be the sum of the first

n

terms of the arithmetic sequence

8,

12,

\ldots

and let

s_2

be the sum of the first

n

terms of the arithmetic sequence

17,

19,

\ldots.

Then

s_1=s_2

for:

no value of

n

one value of

n

two values of

n

four values of

n

more than four values of

n

Write a sum formula for each progression

Cancel the positive factor

n

after equating the sums

The two sums are

\begin{gathered} s_1=2n^2+6n,\\ s_2=n^2+16n. \end{gathered}

Equality gives

n(n-10)=0.

Since a number of terms is positive, only

n=10

works, so there is one value.

Therefore, the correct answer is B.