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

Answer: A

Concepts:arithmetic sequence ratio and proportion

Difficulty rating: 1840

Small Hint:

In an arithmetic sequence, the middle term of the first $21$ terms equals their average

Big Hint:

Express each eleventh term as its corresponding $21$ -term sum divided by $21$

Solution:

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.

1969 AMC 12 Problem 33

Answer: A

Difficulty rating: 1840

Small Hint:

Big Hint:

Solution:

Problem 33 in Other Years