Find the First Term a_n=((n+2)!)/((n+3)!)

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

Step 1

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

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

Step 2

Simplify the factorials.

Add $1$ and $2$ .

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

Add $1$ and $3$ .

$a_{1} = \frac{(3)!}{(4)!}$

Step 3

Rewrite $(4)!$ as $4 \cdot 3!$ .

$a_{1} = \frac{(3)!}{4 \cdot 3!}$

Step 4

Cancel the common factor of $(3)!$ .

Cancel the common factor.

$a_{1} = \frac{(3)!}{4 \cdot 3!}$

Rewrite the expression.

$a_{1} = \frac{1}{4}$