What is the sum of odd place coefficient in the binomial expansion of `( a + b)^(n)`

Options

A. `2^(n)`
B. `2^(n+1)`
C. `2^(n-1)`
D. 2

Answer

To find the sum of the coefficients of the odd-place terms in the binomial expansion of \((a + b)^n\), we can follow these steps:

### Step 1: Write the Binomial Expansion
The binomial expansion of \((a + b)^n\) is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \(\binom{n}{k}\) is the binomial coefficient.

### Step 2: Identify Odd Place Coefficients
In the expansion, the coefficients of the odd-place terms correspond to the terms where \(k\) is odd. These terms are:
- For \(k = 1\): \(\binom{n}{1} a^{n-1} b^1\)
- For \(k = 3\): \(\binom{n}{3} a^{n-3} b^3\)
- For \(k = 5\): \(\binom{n}{5} a^{n-5} b^5\)
- And so on, until \(k = n\) if \(n\) is odd.

### Step 3: Write the Sum of Odd Place Coefficients
The sum of the coefficients of the odd-place terms can be expressed as:
\[ S_{\text{odd}} = \binom{n}{1} + \binom{n}{3} + \binom{n}{5} + \ldots \]

### Step 4: Use the Binomial Theorem for \((a + b)^n\) and \((a - b)^n\)
To find the sum of the coefficients of the odd-place terms, we can use the fact that:
\[ (a + b)^n + (a - b)^n = 2\left(\text{sum of coefficients of even-place terms}\right) \]
\[ (a + b)^n - (a - b)^n = 2\left(\text{sum of coefficients of odd-place terms}\right) \]

### Step 5: Evaluate the Expressions
Let \(x = 1\) and \(y = 1\):
\[ (1 + 1)^n - (1 - 1)^n = 2^n - 0 = 2^n \]
Thus, the sum of the coefficients of the odd-place terms is:
\[ S_{\text{odd}} = \frac{2^n}{2} = 2^{n-1} \]

### Conclusion
The sum of the coefficients of the odd-place terms in the binomial expansion of \((a + b)^n\) is:
\[ S_{\text{odd}} = 2^{n-1} \]