Binomial Theorem

The given series is a type of infinite geometric series. The general form of the series is:

Step 1: Express this as a geometric series with first term and common ratio . The sum of an infinite geometric series is given by: Where is the first term and is the common ratio. Here, and .

Given the expansion: The general term in the expansion is given by: Step 1: Write Equations for the Given Terms From the problem, we know: - 2nd term (when ): - 3rd term (when ): - 4th term (when ): Step 2: Form Ratios for Simplification Dividing the second equation by the first: From binomial coefficients: Thus, Cross-multiplying, Step 3: Form a Second Ratio Dividing the third equation by the second: From binomial coefficients: Thus, From the earlier step, Now substitute this into the second equation: Cross-multiplying: Expanding: Dividing by 2: Using the quadratic formula: Since must be an integer, . Step 4: Compute Since , substituting : Now, Using the given conditions, this simplifies to . Step 5: Final Answer

We are given:

Step 1: Common Denominator
The denominator on the right side is: Thus, Equating the numerators,

Step 2: Expanding Both Terms
Expanding the first term: Expanding the second term: Now combine like terms:

Step 3: Equating Coefficients
By comparing coefficients: - - - -

Step 4: Solving for
Since , substitute this into the second equation: Now from , let and .

Step 5: Calculate

Final Answer: (1)