First slide

Binomial theorem for positive integral Index

Question

In the expansion of ${(x^{2} + 1 + \frac{1}{x^{2}})}^{n}, n \in N,$

Moderate

A

number of terms is 2n + 1
B

constant term is 2^n-1
C

coefficient of x^2n-2 is n
D

coefficient of x² in n

Solution

$(x^{2} + 1 + \frac{1}{x^{2}}) =^{n} C_{0} +^{n} C_{1} (x^{2} + \frac{1}{x^{2}}) +^{n} C_{2} {(x^{2} + \frac{1}{x^{2}})}^{2} + \dots +^{n} C_{n} {(x^{2} + \frac{1}{x^{2}})}^{n}$
This contains terms having $x^{0}, x^{2}, x^{4}, \dots x^{2 n}, x^{- 2}, x^{- 4}, \dots, x^{- 2 n}$
coefficient of constant term $=^{n} C_{0} + (^{n} C_{2}) (2) + (^{n} C_{4}) (^{4} C_{2}) + (^{n} C_{6}) (^{6} C_{3}) + \dots \neq 2^{n - 1}$ .
coefficient of $x^{2 n - 2} {is}^{n} C_{n - 1} = n$
coefficient of $x^{2} {is}^{n} C_{1} + (^{n} C_{3}) (^{3} C_{1}) + (^{n} C_{5}) (^{5} C_{2}) + \dots > n$

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