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 October 14th, 2018, 05:21 AM #1 Newbie   Joined: Oct 2018 From: arizona Posts: 6 Thanks: 0 How Can I Represent These Progressions in Sigma Notation? I would like to represent the following finite progressions in sigma notation: 1. Finding the $n^\text{th}$ term of a geometric progression: $a_n=a_1(r^{n-1})$, where $a_1$ is the first time and $r$ is the common ratio 2. The sum of a geometric progression: $S_n=a_1\frac{1-r^n}{1-r}$ 3. Determining the $n^\text{th}$ term of an arithmetic progression: $a_n=a_1+(n-1)d$, where $d$ is the common difference 4. And finally, the sum of an arithmetic progression: $S_n=\frac{n}{2}(2a_1+(n-1)d)$ Last edited by skipjack; October 14th, 2018 at 06:37 AM.
 October 14th, 2018, 06:39 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,089 Thanks: 1902 Sigma notation isn't needed for the first and third parts of the question. 2. $\displaystyle (1 - r)S_n = \sum_{k=1}^n a_1*r^{k-1} - \sum_{k=1}^n a_1*r^{k} = \sum_{k=1}^n a_1*r^{k-1} - \sum_{k=2}^{n+1} a_1*r^{k-1} = a_1- a_1r^n$ $\displaystyle \therefore S_n = a_1\frac{1 - r^n}{1 - r}$ 4. $\displaystyle 2S_n = \sum_{k=1}^n (a_1 + (k - 1)d) + \sum_{k=1}^n (a_1 + (n - k)d) = \sum_{k=1}^n (2a_1 + (n - 1)d) = n(2a_1 + (n - 1)d)$ $\displaystyle \therefore S_n = {\small\frac{n}{2}}(2a_1 + (n - 1)d)$

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