Arithmetic series

​​In a nutshell

An arithmetic series finds the sum of a given number of terms in an arithmetic sequence. 

Calculating the sum of a sequence

To find the sum $$S_n$$ of the first $$n$$ terms of an arithmetic sequence, you can use a formula. The sum $$S_n$$ is given using the first term, $$a$$, the value $$n$$, which is the position of the $$n^{\text{th}}$$​ term, and either the common difference, $$d$$, or the last term, $$l$$.​

​$$\boxed {S_n = \dfrac {n}{2} [2a+(n-1)d] \quad or \quad S_n = \dfrac {n}{2} (a+l)}$$​​

Deriving the formula

The terms in any given finite arithmetic sequence are as follows: 

$$a, \ a+d, a+2d,... , a+(n-2)d, \ a +(n-1)d$$​​

This means that the sum of a finite arithmetic sequence will be as follows:

$$S_n=[a]+[a+d]+ [a+2d]+... +[ a+(n-2)d]+ [a +(n-1)d]$$​​​​

Adding two such sums together, expressing one of them in reverse, will give: 

$$\begin{aligned} S_n&= &[a]&+ &[a+d] &+... &+&[ a+(n-2)d]&+ &[a +(n-1)d] \\ S_n&= &[a+(n-1)d]&+&[a+(n-2)d] &+... &+&[ a+d]&+ &[a ] \\ \hline 2S_n &= &[2a+(n-1)d] &+ &[2a+(n-1)d] &+ ... &+ &[2a+(n-1)d] &+&[2a+(n-1)d] \end{aligned}$$​​

Therefore, since $$n$$ is the total number of terms in the calculation, this will give:

$$2S_n = n(2a+(n-1)d)$$​​

$$\boxed{S_n = \dfrac {n}2 (2a+(n-1)d)}$$​​

If $$n$$ represents the total number of terms, then the last term is $$l = a+(n-1)d$$. Hence an alternative formula can be found:

$$S_n = \dfrac {n}2 (a+[a+(n-1)d])$$​​

​$$\boxed{S_n = \dfrac {n}2 (a+l) }$$​​

Example 1

Find the sum of the first $$25$$ terms of the arithmetic sequence with a first term of $$8$$, and a common difference of $$3$$. 

You can identify the first few terms of the sequence:

​$$8,11,14,17,...$$​​

To sum the first fifteen terms, without actually finding them and adding them, identify the formula to use:

$$S_n = \dfrac{n}2 [2a + (n-1)d]$$​​

​

Substitute values to find the sum: 

$$S_{25} = \dfrac {25}{2} [2(8) + ( 25-1)3]$$​​

$$\underline{S_{25} = 1100}$$​​

Example 2

Given that the sum of the first $$15$$ terms of an arithmetic sequence is $$630$$ and the last term is $$76$$, what is the first term of this sequence?

Identify the relevant formula to use:

$$S_n = \dfrac {n}{2} (a+l)$$​​

Substitute the given values and solve for $$a$$:​

$$S_{15} = \dfrac{15}{2}(a+76)$$​​

​$$630 = \dfrac{15}{2}(a+76)$$​​

​$$84 = a + 76$$

​$$\underline{a=8}$$​​