To find the [tex]$17^{\text{th}}$[/tex] term of the sequence given by [tex]$t_n = \frac{n(n+3)}{n+1}$[/tex], follow these steps:
1. Understand the formula: The sequence is defined by the term
[tex]\[
t_n = \frac{n(n+3)}{n+1}.
\][/tex]
2. Substitute [tex]$n = 17$[/tex] into the formula: To find the [tex]$17^{\text{th}}$[/tex] term, replace [tex]$n$[/tex] with 17.
[tex]\[
t_{17} = \frac{17(17+3)}{17+1}.
\][/tex]
3. Simplify the expression inside the parentheses: Calculate [tex]$17 + 3$[/tex] and [tex]$17 + 1$[/tex].
[tex]\[
t_{17} = \frac{17 \cdot 20}{18}.
\][/tex]
4. Multiply the terms in the numerator: Compute [tex]$17 \times 20$[/tex].
[tex]\[
t_{17} = \frac{340}{18}.
\][/tex]
5. Divide the numerator by the denominator: Perform the division.
[tex]\[
t_{17} = \frac{340}{18} = 18.88888888888889.
\][/tex]
Therefore, the [tex]$17^{\text{th}}$[/tex] term of the sequence is
[tex]\[
t_{17} = 18.88888888888889.
\][/tex]
