First slide

Binomial theorem for positive integral Index

Question

If the middle term of ${(x^{2} + \frac{1}{x})}^{n}$ is $924 x^{6},$ then value of $n$ is

Moderate

A

8
B

10
C

12
D

20

Solution

As there is just one middle term, $n$ must be even and
middle term is $(\frac{n}{2} + 1)$ th term, and it is given by
$\begin{array}{l} T_{n / 2 + 1} =^{n} C_{n / 2} {(x^{2})}^{n - n / 2} {(\frac{1}{x})}^{n / 2} \\ =^{n} C_{n / 2} x^{n / 2} \end{array}$
Now, $^{n} C_{n / 2} x^{n / 2} = 924 x^{6}$
$\to n / 2 = 6 \to n = 12$
[Also, note that $^{12} C_{6} = 924]$

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