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…

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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______

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{d} ightarrow 	ext { common diff. } \ & \mathrm{A}_{\mathrm{k}}=-\mathrm{kd}[2 \mathrm{a}+(2 \mathrm{k}-1) \mathrm{d}] \ & \mathrm{A}_3=-153 \ & \Rightarrow 153=13 \mathrm{~d}[2 \mathrm{a}+5 \mathrm{~d}] \ & 51=\mathrm{d}[2 \mathrm{a}+5 \mathrm{~d}]...(1) \ & \mathrm{A}_5=-435 \ & 435=5 \mathrm{~d}[2 \mathrm{a}+9 \mathrm{~d}] \ & 87=\mathrm{d}[2 \mathrm{a}+9 \mathrm{~d}] \ & (2)-(1) \ & 36=4 \mathrm{~d}^2 \ & \mathrm{~d}=3, \mathrm{a}=1 \ & \mathrm{a}_{17}-\mathrm{A}_7=49-[-7.3[2+39]]=910\end{aligned}$

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let…

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