Question

If $C_n$ is the coefficient of $x^n$ in the expansion of $(1+x{)}^n$ then $C_1+2.C_2+3.C_3+\dots +n.C_n=$

• A. $2^n$
• B. $n{.2}^n$
• C. $n{.2}^{n+1}$
• D. $n{.2}^{n-1}$

Answer 1

The correct option is D $n{.2}^{n-1}$

$(1+x^n)={\sum }_{r=0}^n{C}_r{x}^n=$ ${}^n{C}_0{x}^0$+${}^n{C}_{1}x$+${}^n{C}_2{x}^2$+.....+${}^n{C}_n{x}^n$

Now,$(1+x^n)=C_0+C_{1}x+C_2{x}^2+.....+C_n{x}^n$ ..(1)

Differentiating (1) w.r.t $x$

$n{(1+x)}^{n-1}=C_1+{2C}_{2}x+.....+{nC}_n{x}^{n-1}$ ..(2)

put $x=1$ and get,

$n{2}^{n-1}=C_1+{2C}_2+.....+{nC}_n$

$C_1+{2C}_2+.....+{nC}_n=n{.2}^{n-1}$