JEE Main 2022 · 30 Jul · Shift 2 · Mathematics Q67

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MathematicsSingle correct+4 / -1Question 67 of 90

For |α| ≥ 1, let $5^{4-2\alpha}$ , 63, $5^{2\alpha-1}$ be in A.P. with the common difference d. Let S $_n$ denote the sum of the first n terms of an A.P., whose first term is α $^2$ and the common difference is d. If S $_{30}$ − S $_{15}$ = 30k, then k is equal to

715

695

684

683

Answer and worked solution

Official answer: C

Find α. $5^{4-2\alpha}+5^{2\alpha-1}=126$ ; with $t=5^{2\alpha}$ , $t^2-630t+3125=0$ , so $t=625$ ( $\alpha=2$ ) or $t=5$ ( $\alpha=\frac12$ , rejected). The A.P. 1, 63, 125 has $d=62$ .

Sum of terms 16 to 30 with first term 4: $15\times4+62(15+16+\dots+29)=60+62\times330=20520=30k$ , so $k=684$ .

(C)

Topics:Arithmetic progression Logarithms

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JEE Main 2022 · 30 Jul · Shift 2, Mathematics question 67