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The expansion of  \left(x+\frac{1}{x}\right)^{n} is

Option: 1

\sum_{r=0}^{n}^{n} C_{r} \;x^{n-2 r}


Option: 2

\sum_{r=0}^{n} ^n C_{r}\; x^{2 n-r}


Option: 3

\sum_{r=0}^{n} (^n C_r x^{ n - r})


Option: 4

None of these


Answers (1)

best_answer

Binomial Theorem\begin{aligned}(x+y)^{n} &=\sum_{r=0}^{n}\left(\begin{array}{l}{n} \\ {r}\end{array}\right) x^{n-r} y^{r} \\ &=x^{n}+\left(\begin{array}{l}{n} \\ {1}\end{array}\right) x^{n-1} y+\left(\begin{array}{c}{n} \\ {2}\end{array}\right) x^{n-2} y^{2}+\ldots+\left(\begin{array}{c}{n} \\ {n-1}\end{array}\right) x y^{n-1}+y^{n} \end{aligned}

 

Now,

\begin{aligned}\left(x+\frac{1}{x}\right)^{n} &=\sum_{r=0}^{n}\;^nC_r\; x^{n-r} \cdot\left(\frac{1}{x}\right)^{r} \\ &=\sum_{r=0}^{n} \;^nC_r\; x^{n-r} x^{-r} \\ &=\sum_{r=0}^{n}\;^nC_r\; x^{n-2 r} \end{aligned}

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HARSH KANKARIA

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