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Factorial Pattern

$$1\times1!=2!-1$$ $$1\times1!+2\times2!=3!-1$$ $$1\times1!+2\times2!+3\times3!=4!-1$$
Does this pattern continue?

Square Factorials

Source: Woody at Maths Jam
Multiply together the first 100 factorials:
$$1!\times2!\times3!\times...\times100!$$
Find a number, $$n$$, such that dividing this product by $$n!$$ produces a square number.

17 December

In March, I posted the puzzle One Hundred Factorial, which asked how many zeros 100! ends with.
What is the smallest number, n, such that n! ends with 50 zeros?

One Hundred Factorial

How many zeros does $$100!$$ end with?