Summing a finite series of natural numbers

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Really we'll only treat the case starting with 1, but this can be easily extended to cover starting at some n0<N. Given a sequence of numbers 1,2,3,,N, the series is given by 1+2+3++N. Finding the sum of the series is actually quite simple:

Given the series:

i=1Ni=1+2+3++N

We can rearrange the series like so:

i=1Ni=(1+N)+(2+N1)+=(N+1)+(N+1)+

The question is where does it stop? That is, how many N+1 terms do we have? Obviously if there is an even number of elements in the sequence you can do this exactly N/2 times. This gives us:

i=1Ni=N(N+1)2

But what if its odd? Well, the middle term will now be (N+1)/2 (think about that a minute and it will be obvious to you), but you only have N+1 repeated (N1)/2 times. That gives us the following

i=1Ni=(N+1)(N1)2+N+12=N(N+1)2

So we see that this series always converges to the same formula.

In the above case, we have seen that N represents the largest number in the sequence. If we are looking at a complete graph G(V,E) where N represents the number of nodes (N=|V|), the sequence goes from 1 to N1. How will this change answer? How would it change your answer if we started at 0 and went up to N?

SummingFiniteIntegerSeries (last edited 2020-01-26 21:02:57 by 68)