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 Mathmatizer July 11th, 2018 03:35 PM

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?)

 romsek July 11th, 2018 03:52 PM

\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.

 Mathmatizer July 12th, 2018 08:24 AM

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!

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