1.

The value of `1^2.C_1 ``+ 3^2.C_3 +`` 5^2.C_5 + ...` isA. `n(n -1)2^(n-2) + n.2^(n-1)`B. `n(n -1)2^(n-2)`C. `n(n+1). 2 ^(n-3)`D. none of these

Answer» Correct Answer - c
We have,
`sum_(r=1) ^(n) r^(2) ""^(n)C_(r) = n (n-1)2^(n-2) + n.2 ^(n-1)`
and , `sum_(r=1)^(n) (-1)^(r-1) r^(2) ""^(n)C_(r) = 0`
Adding these two, 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)`
`rArr 1^(2) C_(1) + 3^(2) . C_(3) + 5^(2). C_(5) +...= n(n -1) 2^(n-3) + n. 2^(n -2)` .


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