How can we prove that the sum of a geometric series is of the order of the last term for r>1, that is to show that its theta notation consists of the last term.
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For $a=1$ the statement is $$na=O(n)$$ which is true.
Otherwese, $$a+ar+ar^2+...+ar^n = \frac {a(r^{n+1}-1)}{r-1} =O(r^n)$$
Mohammad Riazi-Kermani
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