The Sum of a Finite Arithmetic Sequence

When the terms of an arithmetic sequence are added together, an arithmetic series is formed.

Arithmetic Series

Suppose a sequence of numbers is arithmetic and you want to find the sum of the first n terms. There are two different equations that can be used.

(1) $S_n= \frac n 2 (a+l)$
Where:
$S_n$ refers to the sum of the first $n$ terms in a series
$n$ refers to the number of terms in an arithmetic series
$a$ refers to the first term in the series
$l$ refers to the last term

(2) $S_n=\frac n 2 [2a+(n-1)d]$
Where:
$S_n$ refers to the sum of the first $n$ terms in a series
$n$ refers to the number of terms in an arithmetic series
$a$ refers to the first term in the series
$d$ refers to the common difference

So which equation do you use?
It depends on the information provided in the question.

If you have or need to calculate the value of $l$ then you will use the first equation.

If you have or need to calculate the value of $d$ then you will use the second equation.

Example 1

The first term of a sequence is 14 and the sum of the first 15 terms is 975.
Find the 15th term.

The information in the question tells us that:
$a=14$
$n=15$
$S_{15} =975$
$l=$ the 15th term

Given this information, we must use the first formula to calculate what the 15th term is.

$S_n= \frac n 2 (a+l)$ where $S_{15}= \frac {15}{2} (14+l)=975$
$975= \frac {15}{2} (14+l)$
$7.5(14+l)=975$
$14+l=\frac {975}{7.5}$
$l=130-14$
$l=116$

The 15th term is 116.

Example 2

The first term of an arithmetic sequence is -5 and the sixth term is 30.
What is the sum of the first 10 terms of the sequence?

From the question we know:
$a=-5$
$t_n=a+(n-1)d$
$t_6=-5+(6-1)d$
$t_6=30$

We now need to find what the value of $d$ is:
$-5+5d=30$
$5d=35$
$d=7$

Now that we know what $d$ is, we can use the formula to answer the question.

The values that we have are:
$a=-5$
$n=10$
$d=7$

We will therefore use the second formula.

$S_n=\frac n 2 [2a+(n-1)d]$
$S_{10}=\frac {10}{2} [2(-5)+(10-1)7]$
$S_{10}=5 [-10+63]$
$S_{10}=5 \times 53$
$S_{10}=265$

Where $a=-5$ and $d=7$ , the sum of the first ten terms in 265.

Example 3

Find the sum of the first six given terms of an arithmetic sequence

$8,12,16,20,24,28$

The information in the question tells us that:
$a=8$
$l=28$
$n=6$
$d=4$

Given this, we could use either formula as we have all of the information for both.

$S_n=\frac n 2 (a+l)$
$S_6=\frac 6 2 (8+28)$
$S_6=3 \times 36$
$S_6=108$

$S_n=\frac n 2 [2a+(n-1)d]$
$S_6=\frac 6 2 [2(8)+(6-1)4]$
$S_6= 3 [16+(5)4]$
$S_6= 3 \times 36$
$S_6=108$

As you get the same result for both formulas, it is up to you which to use!

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