1. Tardigrade
  2. Question
  3. Mathematics
  4. The number of distinct terms in the expansion of (x3+1+(1/x3))n ; x ∈ R+ and n ∈ N is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Binomial Theorem

Solution:

$\left(x^{3}+1+\frac{1}{x^{3}}\right)^{n}=\left[1+\left(x^{3}+\frac{1}{x^{3}}\right)\right]^{n}$
$={ }^{n} C_{0}+{ }^{n} C_{1}\left(x^{3}+\frac{1}{x^{3}}\right)+\ldots+{ }^{n} C_{n}\left(x^{3}+\frac{1}{x^{3}}\right)^{n}$
All the terms are distinct with powers $\left(x^{3}\right)^{0},\left(x^{3}\right),\left(x^{3}\right)^{2}, \ldots\left(x^{3}\right){ }^{n},\left(x^{3}\right)^{-n}, \ldots\left(x^{3}\right)^{-1}$.
Hence, $(2 n+1)$ terms.