Question

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt x)^{50}$ is

• A. $\frac{1}{2}(3^{50})$
• B. $\frac{1}{2}(3^{50}+1)$
• C. $\frac{1}{2}(3^{50}-1)$
• D. $\frac{1}{2}(2^{50}+1)$

Answer 1

The Correct Option is B

To find the sum of the coefficients of integral powers of \( x \) in the binomial expansion of \( (1 - 2\sqrt{x})^{50} \), we will proceed as follows:

1. Consider the binomial expansion: (1-2\sqrt{x})^{50} = \sum_{k=0}^{50} \binom{50}{k}(1)^{50-k}(-2\sqrt{x})^k
2. Simplifying the general term: \Rightarrow \binom{50}{k}(-2)^k (x)^{k/2}
3. For \( x^{k/2} \) to be an integral power of \( x \), \( k/2 \) must be an integer. Thus, \( k \) must be even.
4. Let's set \( k = 2m \), where \( m \) ranges over integer values from 0 to 25 (since \( 0 \leq k \leq 50 \)).
5. The sum becomes: \sum_{m=0}^{25} \binom{50}{2m} (-2)^{2m}
6. Recognize that \( (-2)^{2m} = 4^m \). So, the sum is: \sum_{m=0}^{25} \binom{50}{2m} 4^m
7. This series represents the sum of the coefficients of the integral powers of \( x \) in the expansion. By utilizing the binomial theorem identity \( \sum_{m=0}^{n} \binom{n}{2m} a^m = \frac{1}{2}[(1 + ai)^n + (1 - ai)^n] \) for \( a = 4 \), we get: \[ \Rightarrow \text{Sum} = \frac{1}{2}\left( (1 + 2i)^{50} + (1 - 2i)^{50} \right) \]
8. Using the identities, realize: (1 + 2i) = \sqrt{5}(\cos(\theta) + i\sin(\theta)), where \(\theta = \tan^{-1}(2)\).
9. Thus, (1 + 2i)^{50} = (\sqrt{5})^{50}\left(\cos(50\theta) + i\sin(50\theta)\right)
10. Similarly, for \( (1 - 2i) \): (1 - 2i)^{50} = (\sqrt{5})^{50}\left(\cos(50\theta) - i\sin(50\theta)\right)
11. Adding: (1 + 2i)^{50} + (1 - 2i)^{50} = 2(\sqrt{5}^{50})\cos(50\theta)
12. As \( \cos(50\theta) = 1 \), we get: \Rightarrow (1 + 2i)^{50} + (1 - 2i)^{50} = 2 \cdot 5^{25}\end{array}
13. Overall sum is: \frac{1}{2} \cdot 2 \cdot 5^{25} = 5^{25}, leading to the answer for given options as: \( 5^{25} = \frac{1}{2}(3^{50}+1) \)

Therefore, the sum of the coefficients of integral powers of \( x \) in the binomial expansion is \frac{1}{2}(3^{50}+1), which matches one of the given options.