Question

If n is even and the middle term in the expansion of \((x^2+\frac{1}{x})^n\) is 924 x⁶, then n is equal to

• A. 12
• B. 10
• C. 8
• D. 14

Hint:

In binomial expansions, the general term for \( \left(a + b\right)^n \) is given by \( T_k = \binom{n}{k} a^{n-k} b^k \). For terms involving powers of \(x\), carefully equate the exponents to find the term corresponding to the given power of \(x\). This is especially useful when determining the middle term in the expansion.

Answer 1

The Correct Option is A

The correct answer is (A) :12 .

Answer 2

The correct answer is: (A) 12.

We are given the expansion of \( \left(x^2 + \frac{1}{x}\right)^n \) and the middle term is 924 \( x^6 \).

In the binomial expansion, the general term is given by:

\( T_k = \binom{n}{k} \left(x^2\right)^{n-k} \left(\frac{1}{x}\right)^k \)

Simplifying the powers of \(x\), we get:

\( T_k = \binom{n}{k} x^{2(n-k)} x^{-k} = \binom{n}{k} x^{2n-3k} \)

We are told that the middle term corresponds to \(x^6\). Since the power of \(x\) in the middle term is \(6\), we set the exponent equal to 6:

\( 2n - 3k = 6 \)

This is the equation we need to solve. Given that \(n\) is even, the middle term occurs at \(k = \frac{n}{2}\). Substituting \(k = \frac{n}{2}\) into the equation:

\( 2n - 3\left(\frac{n}{2}\right) = 6 \)

Simplifying:

\( 2n - \frac{3n}{2} = 6 \)

\( \frac{4n}{2} - \frac{3n}{2} = 6 \)

\( \frac{n}{2} = 6 \)

Solving for \(n\):

\( n = 12 \)

Therefore, the value of \(n\) is (A) 12.