Question

Find the sum of n terms of the series $S_n=1\ast 2+3\ast 2^2+5\ast 2^3+9\ast 2^4+.......$

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

$S_n=1\ast 2+3\ast 2^2+5\ast 2^3+9\ast 2^4+.......$
$=2+(2+1)2^2+(2^2+1)2^3+(2^3+1)2^4+........$
$=(2+2^3+2^5+2^7+.....+2^{2n-1})+(2^1+2^2+2^3+2^4+2^5+............+2^n)-2$
$=2(2^{2n}-1)/(2^2-1)+2(2^n-1)/(2-1)-2$
$=2(2^{2n}-1)/3+2^{n+1}–2-2$
$=\frac{(2^{2n+1}-2+3\ast 2^{n+1}-12)}{3}=\frac{(2^{2n+1}+3\ast 2^{n+1}-14)}{3}$