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  • The number of distinct terms in the expansion of  <img alt="\small \left(x^3+1+\frac{1}{x^3}\right)^n ; x \in R^{+}\ and\ n \in N" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

The number of distinct terms in the expansion of  \small \left(x^3+1+\frac{1}{x^3}\right)^n ; x \in R^{+}\ and\ n \in N is ....

Option: 1

2n


Option: 2

2n+1


Option: 3

2n-1


Option: 4

2n+2


Answers (1)

best_answer

\begin{aligned} & \left(x^3+1+\frac{1}{x^3}\right)^n=\left\{1+\left(x^3+\frac{1}{x^3}\right)\right\}^n \\ & =1+{ }^n C_1\left(x^3+\frac{1}{x^3}\right)^1+{ }^n C_2\left(x^3+\frac{1}{x^3}\right)^2+\ldots+{ }^n C_n\left(x^3+\frac{1}{x^3}\right)^n \end{aligned}

All the terms are distinct with powers

\left(x^3\right)^0,\left(x^3\right)^1,\left(x^3\right)^2, \ldots,\left(x^3\right)^n,\left(x^3\right)^{-n},\left(x^3\right)^{-n+1}, \ldots,\left(x^3\right)^{-1}

Hence, total terms =2 n+1

Posted by

Kuldeep Maurya

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