# sequences - monotonic or not

by rohan03
Tags: monotonic, sequences
 P: 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.
 P: 748 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.
 P: 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
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P: 15,673

## sequences - monotonic or not

What is the definition of n! for you?
 P: 56 well n! means = any number say 5 then multiplied by 5x4x3x2x1 ( natural numbers in hughest to lowest order)
 P: 56 so basically product of positive integres less than or equal to n
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 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 $$(n+1)!=(n+1)*n!$$
 P: 56 thank you this makes sense- sometimes I just get frustrated with not enough explaination at beiggners level

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