In the expansion of `(1+x)^(70)`, the sum of coefficients of odd powers of `x` is

Options

A. `0`
B. `2^(69)`
C. `2^(70)`
D. `2^(71)`

Explanation

To find the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{70} \), 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 Even and Odd Powers
In the expansion, the coefficients of odd powers of \( x \) correspond to the terms where \( k \) is odd, and the coefficients of even powers correspond to the terms where \( k \) is even.

### Step 3: Use the Property of Coefficients
It is known that the sum of the coefficients of odd powers of \( x \) is equal to the sum of the coefficients of even powers of \( x \) in the expansion of \( (1+x)^n \). Therefore, we can denote:
- \( S_{\text{odd}} \): Sum of coefficients of odd powers
- \( S_{\text{even}} \): Sum of coefficients of even powers

From the property, we have:
\[ S_{\text{odd}} = S_{\text{even}} \]

### Step 4: Calculate Total Sum of Coefficients
The total sum of the coefficients in the expansion of \( (1+x)^{70} \) can be found by substituting \( x = 1 \):
\[ (1+1)^{70} = 2^{70} \]
This total sum can be expressed as:
\[ S_{\text{odd}} + S_{\text{even}} = 2^{70} \]

### Step 5: Set Up the Equation
Since \( S_{\text{odd}} = S_{\text{even}} \), we can denote \( S_{\text{odd}} = S_{\text{even}} = S \). Thus, we have:
\[ S + S = 2^{70} \]
or
\[ 2S = 2^{70} \]

### Step 6: Solve for \( S \)
Dividing both sides by 2 gives:
\[ S = \frac{2^{70}}{2} = 2^{69} \]

### Conclusion
Thus, the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{70} \) is:
\[ \boxed{2^{69}} \]

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