Recurrence formula for a sequence

For sequence $\{a_{n}\}$ , $a_{1}=\frac{3}{2}$ , for all $n≥2$ , $n∈N^{+}$ , $a_{n}=\frac{3n \cdot a_{n-1}}{2a_{n-1}+(n-1)}$ .

(1) Find the explicit formula for $\{a_{n}\}$ ;

(2) Proof: For all $n≥2$ , $n∈N^{+}$ , $\prod_{i=1}^n a_{i}< 2 \cdot n!$ ;

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$