## sequences - monotonic or not

Now I know how this works- but I came across this example and even though I know the answer- the simplification given in the explaination doesn't make sense to me.

the squence is an= {5n/n!}
now applying an+1 and dividing an+1/an
the book indicates = 5/n+1

this is what I don't get how
(5n+1 /(n+1)!)/(5 n/n!) can simplify to that ?

can someone explain please- what am I missing here.
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 We have... $$\frac{5^{n+1}}{(n+1)!}\frac{n!}{5^n} = \frac{5\cdot5^{n}}{(n+1)n!}\frac{n!}{5^n}$$ ...which very easily simplifies to the expression you provided by cancelling out like terms.
 right - this is what is not clear to me- I am very new to pure maths how (n+1)! can be written as - (n+1)n!- may be I am having a dumb moment

Mentor
Blog Entries: 8

## sequences - monotonic or not

What is the definition of n! for you?
 well n! means = any number say 5 then multiplied by 5x4x3x2x1 ( natural numbers in hughest to lowest order)
 so basically product of positive integres less than or equal to n
 Mentor Blog Entries: 8 So, you have $$(n+1)!=(n+1)*n*(n-1)*(n-2)*...*3*2*1$$ Right? But then we have $$(n+1)!=(n+1)*[n*(n-1)*(n-2)*...*3*2*1]$$ And the thing in brackets look familiar, no?? Indeed, the bracketed thing is n! So $$(n+1)!=(n+1)*n!$$
 thank you this makes sense- sometimes I just get frustrated with not enough explaination at beiggners level

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