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

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

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

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

Simplify the factorials.

Add $1$ and $1$ .

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

Add $1$ and $2$ .

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

Rewrite $(3)!$ as $3 \cdot 2!$ .

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

Cancel the common factor of $(2)!$ .

Cancel the common factor.

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

Rewrite the expression.

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