https://file.aiquickdraw.com/mathbot/file/1729750304146ldukshds.jpg

Here are the answers to the questions based on the series provided in the image:

**Q40.** The sum of the series if \( x = -1 \) and \( n \) is an even natural number:

- For even \( n \) and \( x = -1 \), after analyzing the behavior of the series, the correct answer is:  
  **(A) 0**

**Q41.** If the given series can be expanded as \( a_0 x^0 + a_1 x^1 + a_2 x^2 + \ldots + a_{n+1} x^{n+1} \), then the value of the expression \( \frac{a_{n+1}}{a_0} + \frac{a_n}{a_1} + \frac{a_{n-1}}{a_2} + \dots \) up to \( \left( \frac{n}{2} + 1 \right) \) terms is:

- For an even \( n \), the correct answer is:  
  **(C) \( 2^n \left( \frac{n}{2} + 1 \right) \)**

**Q42.** If \( x = 1 \), the sum of the series is:

- After evaluating the series for \( x = 1 \), the correct answer is:  
  **(C) \( 2 \sum_{r=1}^{n} 2^r C_r \)**

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Would you like a detailed step-by-step explanation for any of these answers?

Here are 5 related questions to explore further:
1. How do you expand binomial series for even powers?
2. What is the sum of the binomial coefficients when the exponent is even?
3. How does the value of \( x \) affect the behavior of infinite series?
4. What is the combinatorial interpretation of binomial coefficients \( C_r \)?
5. How can you derive the sum of a geometric series involving binomial coefficients?

**Tip:** When working with binomial series, always verify the behavior of the series at special values like \( x = 1 \) and \( x = -1 \) to simplify the summation.

Solve the series-based problems involving binomial coefficients, provided in the attached image.

Solve the binomial series problems based on the binomial coefficients. Understand key algebraic concepts such as summation of series, binomial expansion, and the role of coefficients. Suitable for high school students in Grades 11-12.

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