The sums of an terms of two arithmetic progressions are in the ratio

Wrong question. It will be 7n + 5 instead of 7n – 5.

Given: Ratio of sum of n terms of 2 AP’s

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

Putting $n=1,2,3 \ldots$ we get $A P_{1}$ as $12,19,26 \ldots$ and $A P_{2}$ as $22,27,32 \ldots$

So, $a_{1}=12, d_{1}=7$ and $a_{2}=22, d_{2}=5$

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

$S_{6}=12+(6-1) 7=47$

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

$S_{6}=22+(6-1) 5=47$

Therefore their $6^{\text {th }}$ terms are equal.

Hence proved.