Writing the Terms of a Sequence Using Factorials

Write the first five terms of the sequence defined by the explicit formula $a_{n} = \frac{5 n}{(n + 2)!} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Substitute $n = 1, n = 2,$ and so on in the formula.
$\begin{array}{l} n = 1 & a_{1} = \frac{5 (1)}{(1 + 2)!} = \frac{5}{3!} = \frac{5}{3 \cdot 2 \cdot 1} = \frac{5}{6} \\ n = 2 & a_{2} = \frac{5 (2)}{(2 + 2)!} = \frac{10}{4!} = \frac{10}{4 \cdot 3 \cdot 2 \cdot 1} = \frac{5}{12} \\ n = 3 & a_{3} = \frac{5 (3)}{(3 + 2)!} = \frac{15}{5!} = \frac{15}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{1}{8} \\ n = 4 & a_{4} = \frac{5 (4)}{(4 + 2)!} = \frac{20}{6!} = \frac{20}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{1}{36} \\ n = 5 & a_{5} = \frac{5 (5)}{(5 + 2)!} = \frac{25}{7!} = \frac{25}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{5}{1, 008} \end{array}$
The first five terms are ${\frac{5}{6}, \frac{5}{12}, \frac{1}{8}, \frac{1}{36}, \frac{5}{1,008}} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write the first five terms of the sequence defined by the explicit formula $a_{n} = \frac{(n + 1)!}{2 n} .$

How did it go?