Middle term in the expansion of `(1+x)^(4n)` is `(AA n in N)`

Options

A. `(2n+1)^(th)`
B. `2n^(th)`
C. `(2n-1)^(th)`
D. None of these

Answer

To find the middle term in the expansion of \((1+x)^{4n}\), we can follow these steps:

### Step 1: Identify the total number of terms in the expansion
The total number of terms in the expansion of \((1+x)^{n}\) is given by \(n + 1\). For our case, \(n = 4n\), so the total number of terms is:
\[ 4n + 1 \]

### Step 2: Determine if \(n\) is even or odd
Since \(n\) is a natural number, \(4n\) is always even. Therefore, we will use the formula for the middle term when \(n\) is even.

### Step 3: Calculate the position of the middle term
When \(n\) is even, the middle term is given by the \(\left(\frac{n}{2} + 1\right)\)th term. In our case:
\[ \text{Middle term position} = \frac{4n}{2} + 1 = 2n + 1 \]

### Step 4: Conclusion
Thus, the middle term in the expansion of \((1+x)^{4n}\) is the \((2n + 1)\)th term.

### Final Answer:
The middle term in the expansion of \((1+x)^{4n}\) is the \((2n + 1)\)th term.

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