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October 26th, 2016, 07:20 AM  #1 
Newbie Joined: Oct 2016 From: UK Posts: 7 Thanks: 0  Geometric Series Formula Proof Question
My textbook says: Sn = u1 +u1r + u1r^(2) ...u1r^(n2) + u1^(n1) rSn = u1r + u1r^(2) + u1r^(3) ...u1r^(n1) +u1^(n) rSn  Sn =u1r^(n)  u1 How can the above statement be true? You can't simplify u1r  u1 or u1r^(2)  u1(r). This would suggest that u1r^(n)  u1r^(n1) = u1r^(n)  u1, which isn't true. Could somebody explain how the statement rSn  Sn is true? Last edited by Tomedb; October 26th, 2016 at 07:26 AM. 
October 26th, 2016, 08:27 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,636 Thanks: 2621 Math Focus: Mainly analysis and algebra 
$$\begin{align*} S_n &= \phantom{}u_1+u_1r+u_1r^2 + \ldots u_1r^{n2} + u_1r^{n1} \\ rS_n &= \phantom{u_1+} u_1r+u_1r^2 + \ldots u_1r^{n2} + u_1r^{n1} + u_1 r^n \\ \text{and now subtract the aligned terms} \qquad rS_nSn &= u_1 \phantom{+u_1r+u_1r^2 + \ldots u_1r^{n2} + u_1r^{n1}}+ u_1 r^n \\ &= u_1r^n  u_1 \end{align*}$$ 
October 26th, 2016, 11:58 AM  #3 
Newbie Joined: Oct 2016 From: UK Posts: 7 Thanks: 0 
Ah, so I tried to subtract vertically adjacent terms, rather than subtracting the sum of the terms. Thank you very much for your help. 
October 26th, 2016, 02:21 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,636 Thanks: 2621 Math Focus: Mainly analysis and algebra 
Your approach can be seen to work if you put successive terms next to each other. You should see a telescoping series.


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