Question

The coefficient of \[{{x}^{n}}\] in the expansion of \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\] is
(a) \[\left( n-1 \right)\]
(b) \[{{\left( -1 \right)}^{n-1}}n\]
(c) \[{{\left( -1 \right)}^{n-1}}{{\left( n-1 \right)}^{2}}\]
(d) \[{{\left( -1 \right)}^{n}}\left( 1-n \right)\]

Answer 1

Hint: Expand the given expression and find the terms of the form \[a{{x}^{n}}\]. Sum the coefficients of all such terms to find the coefficient of \[{{x}^{n}}\] in the expansion of \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\]. Use the fact that \[{{\left( r+1 \right)}^{th}}\] term of the expansion of \[{{\left( a-b \right)}^{n}}\] can be written as \[{}^{n}{{C}_{r}}{{\left( a \right)}^{n-r}}{{\left( -b \right)}^{r}}\].

Complete Step-by-Step solution:
We have the expression \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\]. We have to find the coefficient of \[{{x}^{n}}\] in the given expression. To do so, we will expand the given expression and add the coefficients of all the terms of the form \[a{{x}^{n}}\].
We can rewrite \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\] as \[\left( 1+x \right){{\left( 1-x \right)}^{n}}={{\left( 1-x \right)}^{n}}+x{{\left( 1-x \right)}^{n}}\].
So, we have to find the coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1-x \right)}^{n}}\] and \[x{{\left( 1-x \right)}^{n}}\].
We know that \[{{\left( r+1 \right)}^{th}}\] term of the expansion of \[{{\left( a-b \right)}^{n}}\] can be written as \[{}^{n}{{C}_{r}}{{\left( a \right)}^{n-r}}{{\left( -b \right)}^{r}}\].
We will firstly evaluate the coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1-x \right)}^{n}}\].
Thus, \[{{\left( n+1 \right)}^{th}}\] term of the expansion of \[{{\left( 1-x \right)}^{n}}\] is \[{}^{n}{{C}_{n}}{{\left( 1 \right)}^{n-n}}{{\left( -x \right)}^{n}}\]. This term can be expanded to write as \[{}^{n}{{C}_{n}}{{\left( 1 \right)}^{n-n}}{{\left( -x \right)}^{n}}=\dfrac{n!}{n!0!}1{{\left( -1 \right)}^{n}}{{x}^{n}}={{\left( -1 \right)}^{n}}{{x}^{n}}\].
So, the coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1-x \right)}^{n}}\] is \[{{\left( -1 \right)}^{n}}\].
We will now evaluate the coefficient of \[{{x}^{n}}\] in the expansion of \[x{{\left( 1-x \right)}^{n}}\].
We know that any general term of \[{{\left( 1-x \right)}^{n}}\] can be written as \[{}^{n}{{C}_{r}}{{\left( 1 \right)}^{n-r}}{{\left( -x \right)}^{r}}={}^{n}{{C}_{r}}{{\left( -1 \right)}^{r}}{{x}^{r}}\].
Thus, any general term of \[x{{\left( 1-x \right)}^{n}}\] can be written as \[x\left( {}^{n}{{C}_{r}}{{\left( -1 \right)}^{r}}{{x}^{r}} \right)={}^{n}{{C}_{r}}{{\left( -1 \right)}^{r}}{{x}^{r+1}}\].
We have to find the coefficient of \[{{x}^{n}}\] in the expansion of \[x{{\left( 1-x \right)}^{n}}\].
Thus, we have \[r+1=n\].
\[\Rightarrow r=n-1\]
So, the \[{{\left( r+1 \right)}^{th}}={{\left( n-1+1 \right)}^{th}}={{n}^{th}}\] term of the expansion of \[x{{\left( 1-x \right)}^{n}}\] can be written as \[{}^{n}{{C}_{n-1}}{{\left( -1 \right)}^{n-1}}{{x}^{n}}\].
Thus, the coefficient of \[{{x}^{n}}\] can be written as \[{}^{n}{{C}_{n-1}}{{\left( -1 \right)}^{n-1}}=\dfrac{n!}{\left( n-1 \right)!1!}{{\left( -1 \right)}^{n-1}}=\dfrac{n\times \left( n-1 \right)!}{\left( n-1 \right)!}{{\left( -1 \right)}^{n-1}}={{\left( -1 \right)}^{n-1}}n\].
We observe that coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1-x \right)}^{n}}\] is \[{{\left( -1 \right)}^{n}}\] and \[x{{\left( 1-x \right)}^{n}}\] is \[{{\left( -1 \right)}^{n-1}}n\]. No other terms have the term \[{{x}^{n}}\]. We will now add these two coefficients of \[{{x}^{n}}\].
Thus, the coefficient of \[{{x}^{n}}\] in the expansion of \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\] is \[{{\left( -1 \right)}^{n}}+{{\left( -1 \right)}^{n-1}}n={{\left( -1 \right)}^{n}}\{1-n\}={{\left( -1 \right)}^{n}}\left( 1-n \right)\].
Hence, the coefficient of \[{{x}^{n}}\] in the expansion of \[\left( 1+x \right){{\left( 1-x \right)}^{n}}\] is \[{{\left( -1 \right)}^{n}}\left( 1-n \right)\], which is option (d).

Note: To find the coefficient of \[{{x}^{n}}\], it’s not necessary to expand the given expression completely and find the terms. It’s better to simply write the general term of the expression and use it to find the coefficient of \[{{x}^{n}}\]. It’s an easier approach to solve the problem. Also, one must know that \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].