My Math Forum Geometric Series Formula Proof Question

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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^(n-2) + u1^(n-1) rSn = u1r + u1r^(2) + u1r^(3) ...u1r^(n-1) +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^(n-1) = 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: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra \begin{align*} S_n &= \phantom{-}u_1+u_1r+u_1r^2 + \ldots u_1r^{n-2} + u_1r^{n-1} \\ rS_n &= \phantom{-u_1+} u_1r+u_1r^2 + \ldots u_1r^{n-2} + u_1r^{n-1} + u_1 r^n \\ \text{and now subtract the aligned terms} \qquad rS_n-Sn &= -u_1 \phantom{+u_1r+u_1r^2 + \ldots u_1r^{n-2} + u_1r^{n-1}}+ u_1 r^n \\ &= u_1r^n - u_1 \end{align*} Thanks from Tomedb
 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: 7,674 Thanks: 2654 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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