JEE Main 2022 · 26 Jun · Shift 2 · Mathematics Q3

← JEE Main 2022 · 26 Jun · Shift 2

MathematicsSingle correct+4 / -1Question 3 of 90

If $A=\displaystyle\sum_{n=1}^\infty\frac1{\left(3+(-1)^n\right)^n}$ and $B=\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\left(3+(-1)^n\right)^n}$ , then $\dfrac AB$ is equal to :

\dfrac{11}9

-\dfrac{11}9

-\dfrac{11}3

Answer and worked solution

Official answer: C

Split by parity. Odd $n$ : $\frac1{2^n}$ ; even $n$ : $\frac1{4^n}$ .

Sums. $\sum_{\text{odd}}\frac1{2^n}=\frac{1/2}{3/4}=\frac23$ and $\sum_{\text{even}}\frac1{4^n}=\frac{1/16}{15/16}=\frac1{15}$ .

Ratio. $A=\frac23+\frac1{15}=\frac{11}{15}$ , $B=-\frac23+\frac1{15}=-\frac9{15}$ , so $\dfrac AB=-\dfrac{11}9$ .

(C)

Topics:Geometric progression

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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 3