1969 AMC 12 Problem 33

Question

Let

S_n

and

T_n

be the respective sums of the first

n

terms of two arithmetic series. If

S_n:T_n=(7n+1):(4n+27)

for all

n,

the ratio of the eleventh term of the first series to the eleventh term of the second series is:

4:3

3:2

7:4

78:71

undetermined

In an arithmetic sequence, the middle term of the first

21

terms equals their average

Express each eleventh term as its corresponding

21

-term sum divided by

21

For an arithmetic sequence, the eleventh term is the average of the first

21

terms. Thus the respective eleventh terms are

\frac{S_{21}}{21}

and

\frac{T_{21}}{21}.

Their ratio is

\frac{S_{21}}{T_{21}} =\frac{7(21)+1}{4(21)+27} =\frac{148}{111} =\frac43.

Therefore, the correct answer is A.

Problem 33 in other years