Question:
The sums of an terms of two arithmetic progressions are in the ratio (7n – 5) : (5n + 17). Show that their 6th terms are equal.
Solution:
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.
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