If the ratio between the sums of n terms of two arithmetic progressions

Given: Ratio of sum of $\mathrm{n}^{\text {th }}$ terms of 2 AP's

To Find: Ratio of their $11^{\text {th }}$ terms

Let us consider $2 \mathrm{AP}$ series $\mathrm{AP}_{1}$ and $\mathrm{AP}_{2}$.

Putting n = 1, 2, 3… we get AP1 as 8, 15 22… and AP2 as 31, 35, 39….

So, $a_{1}=8, d_{1}=7$ and $a_{2}=31, d_{2}=4$

For $\mathrm{AP}_{1}$

$S_{6}=8+(11-1) 7=87$

For $\mathrm{AP}_{2}$

$S_{6}=31+(11-1) 4=81$

Required ratio $=\frac{87}{81}=\frac{29}{27}$