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

 Let the sum of first n terms of two A.P?s be ${{S}_{n}}$ and ${{S}_{n}}'.$ Then,                $\frac{{{S}_{n}}}{{{S}_{n}}}=\frac{\frac{n}{2}\{2a+(n-1)d\}}{\frac{n}{2}\{2a'+(n-1)d'\}}$ $=\frac{7n+1}{4n+27}$ $\frac{a+\left( \frac{n-1}{2} \right)d}{a'+\left( \frac{n-1}{2} \right)d'}=\frac{7n+1}{4n+27}$                                            ?(i) Also, let ${{m}^{th}}$ term of two A.P?s be ${{T}_{m}}$ and ${{T}_{m}}$? $\frac{{{T}_{m}}}{{{T}_{m}}'}=\frac{a+(m-1)d}{a'+(m-1)d'}$ Replacing $\frac{n-1}{2}$ by $m-1$ in (i), we get $\frac{a+(m-1)d}{a'+(m-1)d'}=\frac{7(2m-1)+1}{4(2m-1)+27}$ $[\because \,n-1=2(m-1)\Rightarrow n=2m-2+1=2m-1]$ $\therefore \frac{{{T}_{m}}}{{{T}_{m}}'}=\frac{14m-7+1}{8m-4+27}=\frac{14m-6}{8m+23}$ $\therefore$  Ratio of ${{m}^{th}}$ term of two A.P's is $14m-6:8m+23$