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
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
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Always check domain conditions after solving equations, especially when \( n \) represents term number.
Updated On: Apr 21, 2026
- \(2 \)
- \(3 \)
- \(4 \)
- \(5 \)
- \(6 \)
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The Correct Option is C
Solution and Explanation
Concept:
Solve the equation \( t_n = 5 \) by simplifying the rational expression.
Step 1: Set equation.
\[ \frac{n(n+6)}{n+4} = 5 \]
Step 2: Cross multiply.
\[ n(n+6) = 5(n+4) \]
Step 3: Simplify.
\[ n^2 + 6n = 5n + 20 \] \[ n^2 + n - 20 = 0 \]
Step 4: Factorize.
\[ (n+5)(n-4)=0 \]
Step 5: Find valid solution.
\[ n = -5, 4 \] Since \( n \in \mathbb{N} \), valid solution is: \[ n = 4 \]
Step 1: Set equation.
\[ \frac{n(n+6)}{n+4} = 5 \]
Step 2: Cross multiply.
\[ n(n+6) = 5(n+4) \]
Step 3: Simplify.
\[ n^2 + 6n = 5n + 20 \] \[ n^2 + n - 20 = 0 \]
Step 4: Factorize.
\[ (n+5)(n-4)=0 \]
Step 5: Find valid solution.
\[ n = -5, 4 \] Since \( n \in \mathbb{N} \), valid solution is: \[ n = 4 \]
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