Question

Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).
Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \] 

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is NOT the correct explanation of (A)
• C. (A) is correct but (R) is false.
• D. (A) is false but (R) is true.

Hint:

The binomial series expansion formula is a powerful tool for approximating expressions. Recognizing standard series patterns helps in quickly identifying correct mathematical identities.

Answer 1

The Correct Option is A

Step 1: Understanding Assertion (A) 
The given series: \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty \] is a known series expansion which simplifies to: \[ \sqrt{4} = 2. \] This means that Assertion (A) is correct. 

Step 2: Understanding Reason (R) 
The Reason (R) states: \[ (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \] This is the binomial expansion for \( (1 - x)^{-n} \), which is a standard result in mathematical series. Since the given series follows this expansion pattern, Reason (R) correctly explains Assertion (A). 

Step 3: Conclusion 
Since both Assertion (A) and Reason (R) are correct and (R) provides a valid explanation for (A), the correct answer is: \[ \text{Both (A) and (R) are correct and (R) is the correct explanation of (A).} \]