C_1 + 2 C_2 + 3 C_3 + 4 C_4 + .... + n C_n = - Vidyadip

MCQ

${C_1} + 2{C_2} + 3{C_3} + 4{C_4} + .... + n{C_n} = $

A
${2^n}$
B
$n.\,\,{2^n}$
✓
$n.\,\,{2^{n - 1}}$
D
$n.\,\,{2^{n + 1}}$

✓

Answer

Correct option: C.

$n.\,\,{2^{n - 1}}$

c
(c) Trick : Put $n = 1,\,\,2,\,\,3,....$

${S_1} = 1,\,\,{S_2} = 2 + 2 = 4$

Now by alternate $(c)$, put $n = 1,\,\,2$

${S_1} = {1.2^0} = 1,{S_2} = {2.2^1} = 4$

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