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

$a_{n} = \frac{n!}{n + 1}$

Step 1

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

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

Step 2

Simplify the numerator.

Expand $(3)!$ to $3 \cdot 2 \cdot 1$ .

$a_{3} = \frac{3 \cdot 2 \cdot 1}{3 + 1}$

Multiply $3 \cdot 2 \cdot 1$ .

Multiply $3$ by $2$ .

$a_{3} = \frac{6 \cdot 1}{3 + 1}$

Multiply $6$ by $1$ .

$a_{3} = \frac{6}{3 + 1}$

Step 3

Reduce the expression by cancelling the common factors.

Add $3$ and $1$ .

$a_{3} = \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

$a_{3} = \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$a_{3} = \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$a_{3} = \frac{3}{2}$