Question

Question #661df

Answer 1

`((n+1)!)/(n+1) = n!" "` (see below, the expression changes!)

Explanation:

This expression can't really be simplified algebraically...

With that said, for fun I'll simplify the expression

`((n+1)!)/(n+1)`

`bb("METHOD 1:"`

In order to simplify the expression

`((n+1)!)/(n+1)`

one thing we could to is set up a table of values by plugging in numbers for `n`:

• `n = 1`:

• `"result" = 1`

• `n = 2`:

• `"result" = 2`

• `n = 3`:

• `"result" = 6`

• `n = 4`:

• `"result" = 24`

• `n = 5`:

• `"result" = 60`

You may be noticing a pattern here: these values yield the same things if the expression were `n!`:

• `1! = 1`

• `2! = 2`

• `3! = 6`

• `4! = 24`

• `5! = 60`

Therefore,

`((n+1)!)/(n+1) = color(blue)(n!`

`" "`

`bb("METHOD 2:"`

We can also use the definition of the factorial, multiplying values decreasing by `1` each time:

`((n+1)!)/(n+1) = (cancel((n+1))(n)(n-1)(n-2)(n-3)(n-4))/(cancel(n+1))`

Notice how after the two terms are canceled, we're left with

`(n)(n-1)(n-2)(n-3)(n-4)`

Which is equivalent to `color(blue)(n!` because each successive term is one integer lower than before it:

`(n)(n-1)(n-2)(n-3)(n-4)... = color(blue)(n!`

Answer 2

It cannot be simplified.

Explanation:

I suspect this question has a typo. If the question was

`((n-1)!)/((n+1)!)`

(note the additional !), then it could be simplified.