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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Sequences and Series

Let a_1, a_2, a_3, be in an arithmetic progression of positive...

Let a1,a2,a3,…a_1, a_2, a_3, \ldots be in an arithmetic progression of positive terms. Let Ak=a12−a22+a32−a42+…+a2k−12−a2k2\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 A3=−153, A5=−435\mathrm{A}_3=-153, \mathrm{~A}_5=-435 and a12+a22+a32=66\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=66, then a17−A7\mathrm{a}_{17}-\mathrm{A}_7 is equal to______

Correct answer

0

Step-by-step solution

d→ common diff. Ak=−kd[2a+(2k−1)d]A3=−153⇒153=13d[2a+5d]51=d[2a+5d]...(1)A5=−435435=5d[2a+9d]87=d[2a+9d](2)−(1)36=4d2d=3,a=1a17−A7=49−[−7.3[2+39]]=910\begin{aligned} & \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}]...(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}
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