Question:

<p><strong>Step 1:</strong> Rewrite the expression as $(1+x)(1-x)^n = (1-x)^n + x(1-x)^n$</p><p><strong>Step 2:</strong> Expand $(1-x)^n$ using binomial theorem: $(1-x)^n = \sum_{r=0}^{n} \binom{n}{r}(-x)^r = \sum_{r=0}^{n} \binom{n}{r}(-1)^r x^r$</p><p><strong>Step 3:</strong> Find coefficient of $x^n$ in $(1-x)^n$: This is $\binom{n}{n}(-1)^n = (-1)^n$</p><p><strong>Step 4:</strong> Find coefficient of $x^n$ in $x(1-x)^n = x\sum_{r=0}^{n} \binom{n}{r}(-1)^r x^r = \sum_{r=0}^{n} \binom{n}{r}(-1)^r x^{r+1}$. For $x^n$ term, we need $r+1=n$, so $r=n-1$. Coefficient is $\binom{n}{n-1}(-1)^{n-1} = n(-1)^{n-1}$</p><p><strong>Step 5:</strong> Total coefficient of $x^n$ is: $(-1)^n + n(-1)^{n-1} = (-1)^n - n(-1)^n = (-1)^n(1-n)$</p><p>∴ Answer: $(-1)^n(1-n)$ or equivalently $(n-1)(-1)^{n+1}$</p>