Question

The general term of a sequence is \( t_n = \frac{n(n+6)}{n+4}, \, n = 1,2,3,\ldots \). If \( t_n = 5 \), then the value of \( n \) is

• A. \(2 \)
• B. \(3 \)
• C. \(4 \)
• D. \(5 \)
• E. \(6 \)

Hint:

Always check domain conditions after solving equations, especially when \( n \) represents term number.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem provides the formula for the n\(^{th}\) term of a sequence and asks to find the specific term number `n` that corresponds to a given value of that term.
Step 2: Key Formula or Approach:
We are given the equation `t_n = 5`. We need to substitute the formula for `t_n` into this equation and solve for `n`.
\[ \frac{n(n+6)}{n+4} = 5 \] Step 3: Detailed Explanation:
Set the given expression for `t_n` equal to 5:
\[ \frac{n(n+6)}{n+4} = 5 \] Multiply both sides by `(n+4)` to eliminate the denominator (assuming `n \neq -4`, which is true since `n` is a positive integer):
\[ n(n+6) = 5(n+4) \] Expand both sides of the equation:
\[ n^2 + 6n = 5n + 20 \] Rearrange the terms to form a standard quadratic equation:
\[ n^2 + 6n - 5n - 20 = 0 \] \[ n^2 + n - 20 = 0 \] Factor the quadratic equation. We need two numbers that multiply to -20 and add to 1. These numbers are 5 and -4.
\[ (n+5)(n-4) = 0 \] This gives two possible solutions for `n`:
\[ n+5 = 0 \quad \text{or} \quad n-4 = 0 \] \[ n = -5 \quad \text{or} \quad n = 4 \] Since `n` represents the term number in a sequence, it must be a positive integer (`n \in \{1, 2, 3, ...\}`). Therefore, we discard the negative solution.
Step 4: Final Answer:
The value of n is 4.