Question

The sum of the series $2. {^{20}C_0} + 5. {^{20}C_1} + 8. {^{20}C_2} + 11. {^{20}C_3} + .... + 62.{^{20}C_{20}}$ is equal to :

• A. $2^{24}$
• B. $2^{25}$
• C. $2^{26}$
• D. $2^{23}$

Answer 1

Answer: (B)

$2.^{20}C_{0} + 5.^{20}C_{1} +8.^{20}C_{2} +11.^{20}C_{3} +.... + 62.^{20}C_{20} $
$ =\sum^{20}_{r=0} \left(3r+2\right)^{20}C_{r} $
$=3 \sum^{20}_{r=0} .{^{20}C_{r}} +2 \sum^{20}_{r=0} {^{20}C_{r}} $
$= 3\sum^{20}_{r=0}r\left(\frac{20}{r}\right)^{19}C_{r-1} +2.2^{20} $
$ = 60.2^{19} +2.2^{20} = 2^{25} $