Question

If \(1\cdot3\cdot5 + 3\cdot5\cdot7 + 5\cdot7\cdot9 + \ldots\) to \(n\) terms is \(n(n+1)f(n)\), then \(f(2) =\)

• A. 12
• B. 42
• C. 18
• D. 20

Hint:

When working with sequences and series, especially those involving patterns or products, it is crucial to derive and verify the general term or sum formula. This ensures accurate calculation and alignment with given terms or conditions.

Answer 1

The Correct Option is D

The sequence given is a series of products of consecutive odd numbers taken three at a time. The general term for the sequence can be expressed as \( (2k-1)(2k+1)(2k+3) \), where \( k \) is the term number starting from 1. To find \( f(2) \), consider the sum of the first two terms of the series: \[ 1\cdot3\cdot5 + 3\cdot5\cdot7 = 15 + 105 = 120 \] Given that this sum is expressed by the formula \( n(n+1)f(n) \) for \( n = 2 \), we have: \[ 2(2+1)f(2) = 120 \quad \Rightarrow \quad 6f(2) = 120 \quad \Rightarrow \quad f(2) = 20 \] Thus, \( f(2) = 20 \), which confirms the correct answer as option (4).