Question

If \[\left| x \right|<1\], then the coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1+x+{{x}^{2}}+{{x}^{3}}+..... \right)}^{2}}\] is
A. n
B. n-1
C. n+2
D. n+1

Answer 1

Hint: In this problem, we have to find the coefficient of \[{{x}^{n}}\] in the expansion of \[{{\left( 1+x+{{x}^{2}}+{{x}^{3}}+..... \right)}^{2}}\]. We should know the binomial theorem formula for negative index to find the coefficient of \[{{x}^{n}}\]. We can first change the given expression into a binomial theorem to find the coefficient of \[{{x}^{n}}\]. We can find the coefficient term in a step by step manner.

Complete step by step solution:
We know that the binomial theorem for negative index 1 is
\[{{\left( 1-x \right)}^{-1}}=1+x+{{x}^{2}}+{{x}^{3}}+.....+{{x}^{n}}\]
We know that the given expansion is,
\[{{\left( 1+x+{{x}^{2}}+{{x}^{3}}+..... \right)}^{2}}\]
We can see that the above binomial theorem formula is similar to the given expression as it has been squared, so we can square the binomial theorem formula to get the given expression value, we get
\[\begin{align}
  & \Rightarrow {{\left( 1+x+{{x}^{2}}+{{x}^{3}}+... \right)}^{2}}={{\left[ {{\left( 1-x \right)}^{-1}} \right]}^{2}} \\
 & \Rightarrow {{\left( 1+x+{{x}^{2}}+{{x}^{3}}+... \right)}^{2}}={{\left( 1-x \right)}^{-2}} \\
\end{align}\]
We also know that next binomial theorem with negative index 2 is,
 \[{{\left( 1-x \right)}^{-2}}=1+2x+3{{x}^{2}}+4{{x}^{3}}+.....+\left( n+1 \right){{x}^{n}}\]
We can see that the above expression has coefficient \[\left( n+1 \right)\] for \[{{x}^{n}}\]in an increasing order.
Therefore, the coefficient of \[{{x}^{n}}\] is \[\left( n+1 \right)\]for the expansion \[{{\left( 1+x+{{x}^{2}}+{{x}^{3}}+..... \right)}^{2}}\].
Therefore, the option D. n+1 is correct and it is the coefficient of \[{{x}^{n}}\].

Note: Students make mistakes while writing the negative index binomial theorem in which we should concentrate. We should always remember the formula for the binomial theorem. In this problem, we have first changed the given expression into a binomial theorem to find the coefficient of \[{{x}^{n}}\].