Question

If $x\in R$, and
$S=1-C_1\frac{1+x}{1+nx}+C_2\frac{1+2x}{{(1+nx)}^2}-C_3\frac{1+3x}{{(1+nx)}^3}+.....upto(n+1)terms$
Then $S$

• A. equals $x^2$
• B. equals $1$
• C. equals $0$
• D. is independent of $x$

Answer 1

Answer: (C), (D)

The correct options are
C equals $0$
D is independent of $x$

$S=1-C_1\frac{1+x}{1+nx}+C_2\frac{1+2x}{{(1+nx)}^2}-C_3\frac{1+3x}{{(1+nx)}^3}+.....upto(n+1)terms$
Putting $\frac{1}{1+nx}=y$, we can write
$S=[C_0-C_{1}y+C_2{y}^2-.....+{(-1)}^n{C}_n{y}^n]-xy[C_1-2C_{2}y+3C_2{y}^2-......+{(-1)}^{n-1}{nC}_n{y}^{n-1}]$
$\Rightarrow S={(1-y)}^n+xy\frac{d}{dt}\left\{{(1-t)}^n\right\}{|}_{t=y}$
$\Rightarrow S={(1-y)}^n-nxy{(1-y)}^{n-1}={(1-y)}^{n-1}[1-y-nxy]$
$\Rightarrow S={(1-y)}^{n-1}[1-(1+nx\left)y\right]=0$
Ans: C,D