Find the Third Term a_n=((n+1)!)/((n+2)!)

$a_{n} = \frac{(n + 1)!}{(n + 2)!}$

Step 1

Substitute in the value of $n$ to find the $n$ th term.

$a_{3} = \frac{((3) + 1)!}{((3) + 2)!}$

Step 2

Simplify the factorials.

Add $3$ and $1$ .

$a_{3} = \frac{(4)!}{((3) + 2)!}$

Add $3$ and $2$ .

$a_{3} = \frac{(4)!}{(5)!}$

Step 3

Rewrite $(5)!$ as $5 \cdot 4!$ .

$a_{3} = \frac{(4)!}{5 \cdot 4!}$

Step 4

Cancel the common factor of $(4)!$ .

Cancel the common factor.

$a_{3} = \frac{(4)!}{5 \cdot 4!}$

Rewrite the expression.

$a_{3} = \frac{1}{5}$