Question

If the ratio of the sum of first n terms of two A.P s is $(7n+1):(4n+27)n$ find the ratio of their mth terms.

Answer 1

Let the sum of first n terms of two A.P's be $S_n$ and $S_n^{\prime }$.

then, $\frac{S_n}{S_n^{\prime }}=\frac{\frac{n}{2}\{2a+(n-1\left)d\right\}}{\frac{n}{2}\{2a^{\prime }+(n-1\left)d^{\prime }\right\}}$

= $\frac{7n+1}{4n+27}$

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

Also, let mth term of two A.P's be $T_m$ and $T_m^{\prime }$

$\frac{T_m}{T_m^{\prime }}=\frac{a+(m-1)d}{a^{\prime }+(m-1)d^{\prime }}$

Replacing $\frac{n-1}{2}$ by m-1 in (i), we get

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

$[\because n-1=2(m-1)\Rightarrow n=2m-2+1=2m-1]$

$\therefore \frac{T_m}{T_m^{\prime }}=\frac{14m-7+1}{8m-4+27}=\frac{14m-6}{8m+23}$

$\therefore$ Ratio of mth term of two A.P's is $14m-6:8m+23$