In the expansion of `(1+x)^30` the sum of the coefficients of odd powers of x is

Options

A. `2^(30)`
B. `2^(31)`
C. 0
D. `2^(29)`

Answer

To find the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{30} \), we can follow these steps:

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

### Step 2: Identify the Coefficients of Odd Powers
In the expansion, the coefficients of the odd powers of \( x \) are \( \binom{30}{1}, \binom{30}{3}, \binom{30}{5}, \ldots \).

### Step 3: Use the Property of Binomial Coefficients
To find the sum of the coefficients of the odd powers, we can use the property:
\[ \text{Sum of coefficients of odd powers} = \frac{(1+1)^{30} - (1-1)^{30}}{2} \]
This is derived from the fact that adding \( (1+x)^{30} \) and \( (1-x)^{30} \) cancels out the odd powers.

### Step 4: Calculate the Values
Now, we calculate:
\[ (1+1)^{30} = 2^{30} \]
and
\[ (1-1)^{30} = 0 \]
Thus, we have:
\[ \text{Sum of coefficients of odd powers} = \frac{2^{30} - 0}{2} = \frac{2^{30}}{2} = 2^{29} \]

### Final Answer
The sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{30} \) is:
\[ \boxed{2^{29}} \]

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