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 July 11th, 2018, 03:35 PM #1 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 Why is the following inequality correct? Why is the following correct: $\displaystyle (c^{n+p-1}+c^{n+p-2}+...+c^{n})d < c^{n}\cdot \frac{1-c^{p}}{1-c}$ where $\displaystyle d=|a_{2}-a_{1}|$ (of series $\displaystyle a_{n}$) **The series $\displaystyle a_{n}$ is part of the problem, but I don't think it has anything to do with this specific inequality, so I didn't provide more details. Only this specific inequality (How to get from left side to right side?)
 July 11th, 2018, 03:52 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,530 Thanks: 1390 \begin{align*} &(c^{n+p-1} + c^{n+p-2} + \dots + c^n) d = \\ \\ &c^n(c^{p-1}+c^{p-2}+\dots +1)d = \\ \\ &c^n d\displaystyle \sum \limits_{k=0}^{p-1}~c^k = \\ \\ &c^n d \dfrac{1-c^p}{1-c} \end{align*} so you'll need some info on $d$ to complete the problem. Thanks from Country Boy and Mathmatizer
 July 12th, 2018, 08:24 AM #3 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 Thank you! $\displaystyle d=|a_{2}-a_{1}|$ and the way the series is defined makes $\displaystyle d = 0.5$ ($\displaystyle a_{2} = 1.5$ and $\displaystyle a_{1}= 1)$ so d<1 that's why the inequality is correct! Thank you! Last edited by Mathmatizer; July 12th, 2018 at 08:26 AM.

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