If the ratio of $k^{ ext {th }}$ terms of two A.P.'s are in the ratio $(2 k-3):(3 k+1)$ and the rat

Question

If the ratio of $k^{	ext {th }}$ terms of two A.P.'s are in the ratio $(2 k-3):(3 k+1)$ and the ratio of their sum to first 65 terms of these two AP's is $\mathrm{a}: \mathrm{b}$ (where a and b are coprime numbers), then the value of $(b+2 a)$ is

Professor PC Thomas Classes · Answer

$$
\frac{A_n}{A_n^{\prime}}=\frac{a_1+(k-1) d_1}{a_2+(k-1) d_2}=\frac{2 k-3}{3 k+1}
$$
put $\mathrm{k}=33$
$$
\begin{aligned}
& \frac{\mathrm{a}_1+32 \mathrm{~d}_1}{\mathrm{a}_2+32 \mathrm{~d}_2}=\frac{2(33)-3}{3(33)+1}=\frac{63}{100} \
& \frac{\mathrm{a}_1+32 \mathrm{~d}_1}{\mathrm{a}_2+32 \mathrm{~d}_2}=\frac{63}{100} \
& \frac{2 \mathrm{a}_1+64 \mathrm{~d}_1}{2 \mathrm{a}_2+64 \mathrm{~d}_2}=\frac{63}{100} \
& \frac{\mathrm{~S}_{65}}{\mathrm{~S}_{65}}=\frac{63}{100} \
& 	herefore \mathrm{a}=63, \mathrm{~b}=100
\end{aligned}
$$