Question

If ${(1+x)}^n={\sum }_{r=0}^n{C}_r{x}^r$ then the value of $3C_1+7C_2+11C^3+....+(4n-1)C_n$ is

• A. $(4n-1)2^n$
• B. $(2n-1)2^n$
• C. $(2n-1)2^n+1$
• D. $(4n-1)2^n-1$

Answer 1

Answer: (C)

$(2n-1)2^n+1$

Let $n=1$, we get
$S_1=3{}^1{C}_1$
$=3$
$=\left(2\right(1)-1)2^1+1$
Let $n=2$ , we get
$S_2=3{}^2{C}_1+7{}^2{C}_2$
$=6+7$
$=13$
$=\left(2\right(2)-1)2^2+1$
Let $n=3$, we get
$S_3=3{}^3{C}_1+7{}^3{C}_2+11{}^3{C}_3$
$=9+21+11$
$=41$
$=\left(2\right(3)-1)2^3+1$
Hence
$S_n$$=\left(2\right(n)-1)2^n+1$