Questions: an = n! / (n+1)! * n^2

Transcript text: $a_{n}=\frac{n!}{(n+1)!} n^{2}$

Solution

Solution Steps

Step 1: Simplify the factorial expression

The given sequence is: \[ a_{n} = \frac{n!}{(n+1)!} n^{2} \] Recall that $ (n+1)! = (n+1) \cdot n! $. Substitute this into the expression: \[ a_{n} = \frac{n!}{(n+1) \cdot n!} n^{2} \]

Step 2: Cancel out $ n! $

The $ n! $ terms in the numerator and denominator cancel out: \[ a_{n} = \frac{1}{n+1} n^{2} \]

Step 3: Simplify the expression

Multiply $ \frac{1}{n+1} $ by $ n^{2} $: \[ a_{n} = \frac{n^{2}}{n+1} \]

Final Answer

$\boxed{a_{n} = \frac{n^{2}}{n+1}}$

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