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Find the number of dissimilar terms in the expansion of $$\bigg(x+\frac{1}{x}+x^2+\frac{1}{x^2} \bigg)^{20}$$

Could someone give me some hint to proceed in this question. I used $x+1/x=t$ and wrote expansion as $(t^2+t-2)^{20}$ but even this does not convey much information. Please help.

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When you square $\left(x^{-2}+x^{-1}+x+x^2\right)$ you get five new terms, namely $x^{-4},x^{-3},1,x^3,x^4$. Thereafter, multiplication by $\left(x^{-2}+x^{-1}+x+x^2\right)$ yields only four new terms.

The cube introduces the four new terms $\left(x^{-4}+x^{-3}+x^3+x^4\right)$, the fourth power introduces the four new terms $\left(x^{-6}+x^{-5}+x^5+x^6\right)$, etc.

So starting with $n=2$ the total number of distinct terms forms an arithmetic sequence

$$ a_n=4n+1 \text{ for } n>1$$

Thus $\left(x^{-2}+x^{-1}+x+x^2\right)^{20}$ will have $4(20)+1=81$ distinct terms.

Note: One could also rewrite it as $\left(\dfrac{1+x+x^3+x^4}{x^2}\right)^{20}$ which would be a degree $80$ polynomial divided by $x^{40}$ which gives a maximum of $81$ terms, but one would have to demonstrate that the missing $x^2$ term does not reduce that number.

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Here are some additional hints.

Applying the binomial theorem we obtain \begin{align*} \left(x+\frac{1}{x}+x^2+\frac{1}{x^2} \right)^{20}=\sum_{j=0}^{20}\binom{20}{j}\left(x+\frac{1}{x}\right)^j\left(x^2+\frac{1}{x^2}\right)^{20-j}\tag{1} \end{align*}

Since \begin{align*} \left(x+\frac{1}{x}\right)\left(x^a+\frac{1}{x^a}\right)=\left(x^{a+1}+\frac{1}{x^{a+1}}\right)+\left(x^{a-1}+\frac{1}{x^{a-1}}\right) \end{align*} and \begin{align*} \left(x^2+\frac{1}{x^2}\right)\left(x^a+\frac{1}{x^a}\right)=\left(x^{a+2}+\frac{1}{x^{a+2}}\right)+\left(x^{a-2}+\frac{1}{x^{a-2}}\right) \end{align*}

we conclude from (1) \begin{align*} \left(x+\frac{1}{x}+x^2+\frac{1}{x^2} \right)^{20}=\sum_{j=0}^{40} a_j\left(x^{j}+\frac{1}{x^j}\right) \end{align*}

Argue that $a_j\ne 0$ where $0\leq j\leq 40$.

It follows the expression has $81$ dissimilar terms.

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  • $\begingroup$ What do you mean by "dissimilar terms"?? $\endgroup$ – Piano Land Mar 19 '18 at 8:52
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    $\begingroup$ @PianoLand: Terms with different powers in $x$. Compare OP's usage of "dissimilar". $\endgroup$ – Markus Scheuer Mar 19 '18 at 9:17

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