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**Difference between 2 "Sum of n terms of geometric series" formulas**

__Notation A)__[tex]S_{n}= \sum_{k=0}^{n - 1} ar^{k} = ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} = \frac{a(1-r^{n})}{1-r}[/tex]

Proof:

[itex] S_{n}= ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1}[/itex]

- [itex] r*S_{n}= ar^{1} + ar^{2} + ar^{3} +...+ ar^{n}[/itex]

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[itex]S_{n} - rS_{n} = a - ar^{n}[/itex]

[itex]S_{n}(1-r)=a(1-r^{n}) \Rightarrow S_{n} = \frac{a(1-r^{n})}{1-r}[/itex]

__Notation B)__[itex]S_{n+1} = \sum_{k=0}^{n} ar^{k} = ar^{0} + ar^{1} + ar^{2} +...+ ar^{n} = \frac{a(1-r^{n+1})}{1-r}[/itex]

Proof:

[itex]S_{n+1} = r^{0} + r^{1} + r^{2} +...+ r^{n}[/itex]

- [itex] r*S_{n+1} = r^{1} + r^{2} + r^{3}+...+ r^{n+1}[/itex]

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[itex]S_{n} - r*S_{n} = r^{0} - r^{n+1}[/itex]

[itex]S_{n}(1-r)=1-r^{n+1} \Rightarrow S_{n} = \frac{1-r^{n+1}}{1-r}[/itex]

So the 2 differences between the formulas I see from the start is:

1) Notation A includes the "a" terms while Notation B only includes the "r" terms. I assume this means that Notation B assumes a=1

2) General terms of [itex]S_{n}[/itex] in Notation A end with exponent [itex]ar^{n-1}[/itex] while General terms of [itex]S_{n+1}[/itex] in Notation B end with exponent [itex]r^{n}[/itex] . I assume the different notation is due to the emphasis put on how the series is described.

If emphasis of notation is placed on [itex]S_{n}[/itex], then you get [itex]ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1}[/itex] thus for the last term, you get the classic formula of "nth term of geometric sequence."

If emphasis of notation is placed on a "cleaner" upper bound on the sigma notation such as [itex]\sum_{n=1}^{n} ar^{k}[/itex] then you get the series in the form [itex]ar^{0} + ar^{1} + ar^{2} +...+ ar^{n}[/itex] so for the last term, you no longer have the classic formula of "nth term of geometric sequence."

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So in summary, what I can conclude when accounting for the differences between the 2 formulas are that the formula will change a little when a=1 and will change again when the exponent of the last term in the series is "n" instead of "n+1."

Are these notations, formulas, and ascertains about the notation differences correct? I'm just making wild guesses, so I'd be appreciative of any feedback that can correct the assumptions I've made, or to add to the accounting for the differences in the notations and formulas.

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