Question

The sums of first n terms of two A.P.'s are in the ratio (7n+2) : (n+4). Find the ratio of their 5th terms.

Answer 1

$\frac{S_n}{S_n^{\prime }}=\frac{7n+2}{n+4}$

$\frac{2a+(n-1)d}{2a^{\prime }+(n-1)d^{\prime }}=\frac{7n+2}{n+4}$

$\frac{a+\left(\frac{n-1}{2}\right)d}{a^{\prime }\left(\frac{n-1}{2}\right)d^{\prime }}=\frac{7n+2}{n+4}$ ………..$\left(1\right)$

$\therefore \frac{a+4d}{a^{\prime }+4d}=\frac{7\times 9+2}{9+4}=\frac{65}{13}=\frac{5}{1}=5:1$

$\frac{T_5}{T_5^{\prime }}=\frac{a+4d}{a^{\prime }+4d^{\prime }}$ ………$\left(2\right)$

On comparing equation $\left(1\right)$ & $\left(2\right)$

$\frac{n-1}{2}=4$

$\therefore n-1=8$

$n=9$.