# 1969 AHSME Problems/Problem 33

## Problem

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: $\text{(A) } 4:3\quad \text{(B) } 3:2\quad \text{(C) } 7:4\quad \text{(D) } 78:71\quad \text{(E) undetermined}$

## Solution

Let $S$ be the first arithmetic sequence and $T$ be the second arithmetic sequence. If $n = 1$, then $S_1:T_1 = 8:31$. Since $S_1$ and $T_1$ are just the first term, the first term of $S$ is $8a$ and the first term of $T$ is $31a$ for some $a$. If $n = 2$, then $S_2:T_2 = 15:35 = 3:7$, so the sum of the first two terms of $S$ is $3b$ and the sum of the first two terms of $T$ is $7b$ for some $b$. Thus, the second term of $S$ is $3b-8a$ and the second term of $T$ is $7b - 31a$, so the common difference of $S$ is $3b-16a$ and the common difference of $T$ is $7b-62a$.

Thus, using the first terms and common differences, the sum of the first three terms of $S$ equals $\tfrac{1}{2} \cdot 3(16a + 2(-16a + 3b))$, and the sum of the first three terms of $T$ equals $\tfrac{1}{2} \cdot 3(62a + 2(-62a + 7b))$. That means $$\frac{16a + 2(-16a + 3b)}{62a + 2(-62a + 7b)} = \frac{22}{39}$$ $$\frac{6b-16a}{14b-62a} = \frac{22}{39}$$ $$\frac{3b-8a}{7b-31a} = \frac{22}{39}$$ $$117b - 312a = 154b - 682a$$ $$-37b = -370a$$ $$b = 10a$$ With the substitution, the common difference of $S$ is $14a$, and the common difference of $T$ is $8a$. That means the $11^\text{th}$ term of $S$ is $8a + 10(14a) = 148a$, and the $11^\text{th}$ term of $T$ is $31a + 10(8a) = 111a$. Thus, the ratio of the eleventh term of the first series to the eleventh term of the second series is $148a:111a = \boxed{\textbf{(A) } 4:3}$.

## Solution 2 (Quick Sol)

Note that the sum of arithmetic sequences can be expressed as a quadratic polynomial of $n$. From the ratio given in the problem, multiply $n$ to both sides to get quadratic polynomials. $$S_n = 7n^2+n, T_n = 4n^2+27n$$ From there, the 11th term for $S_n$ and $T_n$ can becalculated. $$S_{11} - S_{10} = 7*11^2+11 - (7*10^2+10) = 148$$ $$T_{11} - T_{10} = 4*11^2+27*11 - (4*10^2+27*10) = 111$$ The ratio is $148 : 111 = \boxed{\textbf{(A) } 4:3}$.

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