Question

The Coefficient of $x^r$ in the expansion of ${\left(1+x\right)}^{-2}$ is

• A. $r$
• B. $r+1$
• C. $r+3$
• D. $r-1$

Answer 1

The correct option is B

$r+1$

Explanation for the Correct Answer:

Finding the coefficient of $x^r$:

Given Expression is ${\left(1+x\right)}^{-2}$

The given expression is of the form ${\left(1+x\right)}^n$

Hence, the binomial expansion is ${\left(1+x\right)}^n=1+nx+\frac{\left[n\left(n-1\right)\right]}{2!}{x}^2+\dots \dots$

Using the Expansion series we can write,

${\left(1+x\right)}^{-2}=1-2x+3x^2-4x^3+......+(r+1)x^r-...+\infty$

Here, we can see that the coefficient of $x^r$ in the expansion of ${\left(1+x\right)}^{-2}$ is $r+1$

Hence, option (B) is the correct answer.