# Sequences - monotonic or not

1. ### rohan03

56
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.

Last edited: Apr 2, 2012
2. ### Number Nine

779
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.

3. ### rohan03

56
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

Last edited: Apr 2, 2012
4. ### micromass

18,536
Staff Emeritus
What is the definition of n! for you?

5. ### rohan03

56
well n! means = any number say 5 then multiplied by 5x4x3x2x1 ( natural numbers in hughest to lowest order)

6. ### rohan03

56
so basically product of positive integres less than or equal to n

7. ### micromass

18,536
Staff Emeritus
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

$$(n+1)!=(n+1)*n!$$

8. ### rohan03

56
thank you this makes sense- sometimes I just get frustrated with not enough explaination at beiggners level