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October 14th, 2018, 05:21 AM   #1
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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.
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October 14th, 2018, 06:39 AM   #2
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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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