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Q.
If $3^{\text {rd }}, 6^{\text {th }}$ and $10^{\text {th }}$ terms of a non-constant A.P. are in G.P., then the common ratio of G.P. is
Sequences and Series
Solution:
Let first term of A.P. be a and common difference be d, then
$( a +5 d )^2=( a +2 d )( a +9 d )$
$\Rightarrow a ^2+25 d ^2+10 ad = a ^2+11 ad +18 d ^2 $
$\Rightarrow ad =7 d ^2 \Rightarrow a =7 d (\because d \neq 0) $
$\therefore \text { G.P. will be } 9 d , 12 d , 16 d , \ldots \ldots . . $
$\therefore \text { Common difference }=\frac{4}{3} .$