Question:
The $7^{th}$ term in the expansion of $\left(x^{2}+\frac{1}{x^{2}}+2\right)^{n} $ is
The $7^{th}$ term in the expansion of $\left(x^{2}+\frac{1}{x^{2}}+2\right)^{n} $ is
Updated On: Jul 7, 2022
- $\frac{n!}{\left[n/5!\right]^{2}} x^{2}$
- $^nC_6\,x^6$
- $\frac{1.3.5...\left(2n+1\right)}{n!} 2^{n}$
- None of these
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The Correct Option is D
Solution and Explanation
We have, $\left(x^{2}+\frac{1}{x^{2}}+2\right)^{n} = \left(x+\frac{1}{x}\right)^{2n}$
$\therefore T_{7} = T_{6+1} = \,^{2n}C_{6}\left(x\right)^{2n-6} \left(\frac{1}{x}\right)^{6}$
$= \,^{2n}C_{6}\,x^{2n-12}$
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Notes on binomial expansion formula
Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .