Question

If the ratio of the sum of first $n$ terms of two $A.P^{\prime }s$ is $(7n+1):(4n+27)$, find the ratio of their $m^{th}$ terms.

Answer 1

Given, $S_1:S_2=(7n+1):(4n+27)$

$\frac{\frac{n}{2}(2a+(n-1\left)d\right)}{\frac{n}{2}(2a^{\prime }+(n-1\left)d^{\prime }\right)}=\frac{7n+1}{4n+27}$

$\Rightarrow \frac{a+\left(\frac{n-1}{2}\right)d}{a^{\prime }+\left(\frac{n-1}{2}\right)d^{\prime }}=\frac{7n+1}{4n+27}$

We want $\frac{a+(m-1)d}{a^{\prime }+(m-1)d^{\prime }}$

$\Rightarrow m-1=\frac{n-1}{2}$

$\Rightarrow n=2m-2+1=2m-1$

Putting $n=2m-1$

$\frac{a+(m-1)d}{a^{\prime }+(m-1)d^{\prime }}=\frac{7(2m-1)+1}{4(2m-1)+27}=\frac{14m-6}{8m+23}$

$\Rightarrow a_m:a_m^{\prime }=14m-6:8m+23$. ratio of $m^{th}$ terms.