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Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms.
Let $\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1{ }^2-\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2-\mathrm{a}_4{ }^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}{ }^2-\mathrm{a}_{2 \mathrm{k}}{ }^2$.
If $\mathrm{A}_3=-153, \mathrm{~A}_5=-435$ and $\mathrm{a}_1{ }^2+\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2=66$, then $\mathrm{a}_{17}-\mathrm{A}_7$ is equal to....................
$920$
$852$
$910$
$911$
Solution
$ \mathrm{d} \rightarrow \text { common diff. } $
$ \mathrm{A}_{\mathrm{k}}=-\mathrm{kd}[2 \mathrm{a}+(2 \mathrm{k}-1) \mathrm{d}] $
$ \mathrm{A}_3=-153 $
$ \Rightarrow 153=13 \mathrm{~d}[2 \mathrm{a}+5 \mathrm{~d}] $
$ 51=\mathrm{d}[2 \mathrm{a}+5 \mathrm{~d}] $
$ \mathrm{A}_5=-435 $
$ 435=5 \mathrm{~d}[2 \mathrm{a}+9 \mathrm{~d}] $
$ 87=\mathrm{d}[2 \mathrm{a}+9 \mathrm{~d}] $
$ (2)-(1) $
$ 36=4 \mathrm{~d}^2$
$ \mathrm{~d}=3, \mathrm{a}=1 $
$ \mathrm{a}_{17}-\mathrm{A}_7=49-[-7.3[2+39]]=910$
