# 1.1 Sigma notation, finite & Infinite series

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$\begin{array}{|c|}\hline \sum _{i=1}^{n}1=n\\ \hline\end{array}$

Another simple arithmetic sequence is when ${a}_{1}=1$ and $d=1$ , which is the sequence of positive integers:

$\begin{array}{ccc}\hfill {a}_{i}& =& {a}_{1}+d\left(i-1\right)\hfill \\ & =& 1+1·\left(i-1\right)\hfill \\ & =& i\hfill \\ \hfill \left\{{a}_{i}\right\}& =& \left\{1;2;3;4;5;...\right\}\hfill \end{array}$

If we wish to sum this sequence from $i=1$ to any positive integer $n$ , we would write

$\sum _{i=1}^{n}i=1+2+3+...+n$

This is an equation with a very important solution as it gives the answer to the sum of positive integers.

## Interesting fact

Mathematician, Karl Friedrich Gauss, discovered this proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1 to 100. Young Karl realised how to do this almost instantaneously and shocked the teacher with the correct answer, 5050.

We first write ${S}_{n}$ as a sum of terms in ascending order:

${S}_{n}=1+2+...+\left(n-1\right)+n$

We then write the same sum but with the terms in descending order:

${S}_{n}=n+\left(n-1\right)+...+2+1$

We then add corresponding pairs of terms from equations [link] and [link] , and we find that the sum for each pair is the same, $\left(n+1\right)$ :

$2{S}_{n}=\left(n+1\right)+\left(n+1\right)+...+\left(n+1\right)+\left(n+1\right)$

We then have $n$ -number of $\left(n+1\right)$ -terms, and by simplifying we arrive at the final result:

$\begin{array}{ccc}\hfill 2{S}_{n}& =& n\phantom{\rule{0.277778em}{0ex}}\left(n+1\right)\hfill \\ \hfill {S}_{n}& =& \frac{n}{2}\left(n+1\right)\hfill \end{array}$
${S}_{n}=\sum _{i=1}^{n}i=\frac{n}{2}\left(n+1\right)$

Note that this is an example of a quadratic sequence.

## General formula for a finite arithmetic series

If we wish to sum any arithmetic sequence, there is no need to work it out term-for-term. We will now determine the general formula to evaluate a finite arithmetic series. We start with the general formula for an arithmetic sequence and sum it from $i=1$ to any positive integer $n$ :

$\begin{array}{ccc}\hfill \sum _{i=1}^{n}{a}_{i}& =& \sum _{i=1}^{n}\left[{a}_{1}+d\left(i-1\right)\right]\hfill \\ & =& \sum _{i=1}^{n}\left({a}_{1}+di-d\right)\hfill \\ & =& \sum _{i=1}^{n}\left[\left({a}_{1}-d\right)+di\right]\hfill \\ & =& \sum _{i=1}^{n}\left({a}_{1}-d\right)+\sum _{i=1}^{n}\left(di\right)\hfill \\ & =& \sum _{i=1}^{n}\left({a}_{1}-d\right)+d\sum _{i=1}^{n}i\hfill \\ & =& \left({a}_{1}-d\right)n+\frac{dn}{2}\left(n+1\right)\hfill \\ & =& \frac{n}{2}\left(2{a}_{1}-2d+dn+d\right)\hfill \\ & =& \frac{n}{2}\left(2{a}_{1}+dn-d\right)\hfill \\ & =& \frac{n}{2}\left[\phantom{\rule{0.166667em}{0ex}}2{a}_{1}+d\left(n-1\right)\right]\hfill \end{array}$

So, the general formula for determining an arithmetic series is given by

${S}_{n}=\sum _{i=1}^{n}\left[{a}_{1}+d\left(i-1\right)\right]=\frac{n}{2}\left[2{a}_{1}+d\left(n-1\right)\right]$

For example, if we wish to know the series ${S}_{20}$ for the arithmetic sequence ${a}_{i}=3+7\left(i-1\right)$ , we could either calculate each term individually and sum them:

$\begin{array}{ccc}\hfill {S}_{20}& =& \sum _{i=1}^{20}\left[3+7\left(i-1\right)\right]\hfill \\ & =& 3+10+17+24+31+38+45+52+\hfill \\ & & 59+66+73+80+87+94+101+\hfill \\ & & 108+115+122+129+136\hfill \\ & =& 1390\hfill \end{array}$

or, more sensibly, we could use equation [link] noting that ${a}_{1}=3$ , $d=7$ and $n=20$ so that

$\begin{array}{ccc}\hfill {S}_{20}& =& \sum _{i=1}^{20}\left[3+7\left(i-1\right)\right]\hfill \\ & =& \frac{20}{2}\left[2·3+7\left(20-1\right)\right]\hfill \\ & =& 1390\hfill \end{array}$

This example demonstrates how useful equation [link] is.

## Exercises

1. The sum to $n$ terms of an arithmetic series is ${S}_{n}=\frac{n}{2}\left(7n+15\right)$ .
1. How many terms of the series must be added to give a sum of 425?
2. Determine the 6 th term of the series.
2. The sum of an arithmetic series is 100 times its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero.
3. The common difference of an arithmetic series is 3. Calculate the values of $n$ for which the ${n}^{th}$ term of the series is 93, and the sum of the first $n$ terms is 975.
4. The sum of $n$ terms of an arithmetic series is $5{n}^{2}-11n$ for all values of $n$ . Determine the common difference.
5. The sum of an arithmetic series is 100 times the value of its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero.
6. The third term of an arithmetic sequence is -7 and the 7 th term is 9. Determine the sum of the first 51 terms of the sequence.
7. Calculate the sum of the arithmetic series $4+7+10+\cdots +901$ .
8. The common difference of an arithmetic series is 3. Calculate the values of $n$ for which the ${n}^{\mathrm{th}}$ term of the series is 93 and the sum of the first $n$ terms is 975.

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