Question

If \( 1 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 7 + 5 \cdot 7 \cdot 9 + \dots \) (n terms) = \( n(n + 1)f(n) - 3n \), then \( f(1) = \):

• A. \( 9 \)
• B. \( 8 \)
• C. \( 7 \)
• D. \( 6 \)

Hint:

For sum of series problems, analyze the structure of the series and represent it in terms of a general formula. Substitute small values of \(n\) into the given expression to solve for the unknown function.

Answer 1

The Correct Option is A

Step 1: General term in the sequence. The general form of the nth term in the sequence is the product of three consecutive odd numbers, which is given by: \[ T_k = (2k - 1)(2k + 1)(2k + 3). \] Step 2: Relating the sum to the given expression. We are given that: \[ S_n = n(n + 1) f(n) - 3n. \] Thus, we equate the sum to the expression \( n(n + 1) f(n) - 3n \). Step 3: Solving for \( f(1) \). Substituting \( n = 1 \) into the equation: \[ 1 \cdot 3 \cdot 5 = 1(1 + 1) f(1) - 3 \cdot 1. \] \[ 15 = 2 f(1) - 3. \] Solving for \( f(1) \): \[ 18 = 2 f(1), \] \[ f(1) = 9. \] Thus, \( f(1) = 9 \).

Answer 2

Problem: Given the sum \[ 1 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 7 + 5 \cdot 7 \cdot 9 + \dots \quad (n \text{ terms}) = n(n + 1) f(n) - 3n, \] find the value of \( f(1) \).

Step 1: Evaluate the sum for \( n = 1 \) For \( n = 1 \), the sum is just the first term: \[ S_1 = 1 \cdot 3 \cdot 5 = 15. \]

Step 2: Substitute \( n = 1 \) into the given formula \[ S_1 = 1 \times (1 + 1) \times f(1) - 3 \times 1 = 2 f(1) - 3. \]

Step 3: Equate and solve for \( f(1) \) \[ 15 = 2 f(1) - 3, \] \[ 2 f(1) = 18, \] \[ f(1) = 9. \]

Final answer: \[ \boxed{9}. \]