### Brain Teasers

# Rapid Increase

What comes next in this series?

0, 1, 2, 720!, ?

In math, the exclamation mark is a factorial, which is the product of every whole number from 1 to n.

For example, five factorial is written 5! = 1x2x3x4x5 = 120. By definition 0! = 1.

So 720! = 1x2x3x4x...718x719x720, which is a very large number.

0, 1, 2, 720!, ?

In math, the exclamation mark is a factorial, which is the product of every whole number from 1 to n.

For example, five factorial is written 5! = 1x2x3x4x5 = 120. By definition 0! = 1.

So 720! = 1x2x3x4x...718x719x720, which is a very large number.

### Hint

Look at the first few factorials:0! = 1

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

### Answer

The crucial insight is that 6! = 720, so you can rewrite the series as:0, 1, 2, (6!)!, ?

Then note that 3! = 6, so you can rewrite the series again as:

0, 1, 2, ((3!)!)!, ?

You might think the series is:

0!!!, 1!!!, 2!!!, 3!!!, ...

However, 0!!! = 1, so the series must be:

0, 1!, 2!!, 3!!!, ...

Thus, the next term is 4!!!!.

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