Question

The value of $1^2.C_1 + 3^2.C_3 + 5^2.C_5 + ...$ is :

• A. $n (n - 1)^{n -2} + n . 2^{n - 1 }$
• B. $n (n - 1 )^{n - 2 }$
• C. $n (n - 1 )^{n - 3 }$
• D. none of the above

Answer 1

The Correct Option is D

We know $\sum^{n}_{r=1} r^{2} .^{n}C_{r} = n\left(n-1\right)2^{n-2} $ $+ n. 2^{n-1} $ .....(1) and $\sum^{n}_{r=1} \left(-1\right)^{r-1} .r^{2} . ^{n}C_{r} = 0 $ ...(2) Adding (1) & (2) we get $2[1^2 . C_1 + 3^2 . C_3 + 5^2 \, C_5 + ....]$ $= n(n - 1)2^{n-2} + n . 2^{n-1}$ $\Rightarrow \, [1^2 \, C_1 + 3^2 \, C_3 + 5^2 \, C_5 +....]$ $= n(n - 1)2^{n-3 } + n . 2^{n-2}.$