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I just found out (I am an amateur) that if I have the following series I get the following answers for the a nth number . (each series is the sum of the previous one)

$$1 ,1 , 1, \dots, 1 $$

$$1,2,3,4,5, \dots, n/1! $$

$$1,3,6,10,15,21,28,\dots, n(n+1)/2!$$

$$1,4,10,25,\dots, n(n+1)(n+2)/ 3! $$

$$1,5,15,40,\dots, n(n+1)(n+2)(n+3)/4! $$

etc.

I am looking to find a logical explanation to this development

Jean-Claude Arbaut
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Philippe Sussholz
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1 Answers1

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You may have heard of Pascal's Triangle. Each of these sequences fills a sloping line of the triangle.
It can be explaind by induction. For example, $$\frac{n(n+1)(n+2)(n+3)}{24}+\frac{(n+1)(n+2)(n+3)}6=\frac{(n+1)(n+2)(n+3)(n+4)}{24}$$ which shows that adding the fourth series gives the fifth seres.

Empy2
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