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I'm really struggling with the way this book proves the sum of geometric series. As I know the formula of sum of geometric series should be $$s_n= a\frac{1-r^n}{1-r}$$ But the book wrote it differently as $$s_n= a\frac{1-r^{n+1}}{1-r}$$ How is this formula derived?

rubik
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Nothing
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  • First, we usually like in this site to have the formulas written directly and not by means of copy and past, pics or whatever. Second, it is almost impossible to understand what's written in what you pasted. You may want to check the helpful site http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-qu%E2%80%8C%E2%80%8Bick-reference – Timbuc Jan 19 '15 at 18:33
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    The sum of the first $n$ terms is $\frac{a(1-r^n)}{1-r}$. The sum of the terms up to and including $ar^n$ is $\frac{a(1-r^{n+1})}{1-r}$. – André Nicolas Jan 19 '15 at 18:41

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As you have been proven in the text, $$s_n = \sum_{k=0}^n ar^k = a \frac{1-r^{n+1}}{1-r}$$ You just recalled an erroneous formula.

AlexR
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  • Yeah , I know about this , but some of the elementary textbook use the formula S_n = a(1-r^n)/(1-r) , which one is more correct ? – Nothing Jan 19 '15 at 18:38
  • @user208886 There is only one correct formula. Maybe the other book defines $$S_N := \sum_{k=0}^{N-1} ar^k$$ (watch the summation limits) – AlexR Jan 19 '15 at 18:40
  • So means majority of yours learn the formula like in the text I posted just now ? I think I understand it , but I have to know what is the more common formula , thanks – Nothing Jan 19 '15 at 18:45
  • Usually the one from the text, because $S_N$ refers to the partial sum operator, $S_N(a) := \sum_{k=0}^N a_k$ but it doesn't matter at all because they are equivalent. – AlexR Jan 19 '15 at 18:46
  • Oh I see , thanks – Nothing Jan 19 '15 at 18:47
  • Happy to help :) After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark ✓ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?. – AlexR Jan 19 '15 at 18:50
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As I know the formula of sum of geometric series should be $s_n= a(1-r^n)/(1-r)$

Are you sure of this? What's your definition of $s_n$?

Lee
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  • This is what my secondary teacher taught us , he said starting with n = 1,2,3,4.... We should use this formula – Nothing Jan 19 '15 at 18:40
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    It might help you to understand where that formula comes from! Then you would know when it is appropriate to have an $n$ and when it is appropriate to have an $n+1$... – Lee Jan 19 '15 at 18:43