QJEEM22Madb1f: JEE Main 2022 Mathematics - Class 11 JEEM22Madb1f Solution: Let be an A.P. If , then is equal to _________.

Given

$S = {{{a_1}} \over 2} + {{{a_2}} \over {{2^2}}} + {{{a_3}} \over {{2^3}}} + {{{a_4}} \over {{2^4}}}\, + \,.....\,\infty$

${{{1 \over 2}S = {{{a_1}} \over {{2^2}}} + {{{a_2}} \over {{2^3}}}\, + \,.........\,\infty } \over {{S \over 2} = {{{a_1}} \over 2} + {{({a_2} + {a_1})} \over {{2^2}}} + {{({a_3} + {a_2})} \over {{2^3}}}\, + \,......\,\infty }}$

$\Rightarrow {S \over 2} = {{{a_1}} \over 2} + {d \over 2}$

$\Rightarrow {a_1} + d = {a_2} = 4 \Rightarrow 4{a_2} = 16$