Question

What is the value of n when \( t_n = \frac{n(n+6)}{n+4} \) and \( t_n = 5 \)?

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

Hint:

To solve for \( n \) in a given expression, substitute the known value of \( t_n \) and solve the resulting quadratic equation.

Answer 1

The Correct Option is B

We are given the formula for the nth term: \[ t_n = \frac{n(n+6)}{n+4} \] and the value \( t_n = 5 \). We need to solve for \( n \). Substitute \( t_n = 5 \) into the equation: \[ 5 = \frac{n(n+6)}{n+4} \] Multiply both sides by \( (n+4) \) to eliminate the denominator: \[ 5(n+4) = n(n+6) \] Simplifying: \[ 5n + 20 = n^2 + 6n \] Rearrange the equation: \[ n^2 + 6n - 5n - 20 = 0 \] \[ n^2 + n - 20 = 0 \] Solve the quadratic equation using the quadratic formula: \[ n = \frac{-1 \pm \sqrt{1^2 - 4(1)(-20)}}{2(1)} = \frac{-1 \pm \sqrt{81}}{2} \] \[ n = \frac{-1 \pm 9}{2} \] Thus, \( n = 4 \) or \( n = -5 \). Since \( n \) must be positive, the correct value of \( n \) is
4.