Sum of the First \(n\) Terms of an Arithmetic Sequence
(Arithmetic Series)
Given an arithmetic sequence we'll sometimes need to calculate the sum of its first \(n\) terms.
For example, given the arithmetic sequence whose first few terms are:
\[3,7,11,15,19,23, \dots \]
we may need to calculate the sum of its first \(100\) terms.
We could do this by adding one term to the next up to the \(100^{\text{th}}\) term but that would take time.
Instead we use one of two formula.
We have two formula for the sum of the first \(n\) terms of an arithmetic sequence, we learn both in this section.
Formula 1
Given an arithmetic sequence, we can calculathe the sum of its first \(n\) terms, which we write \(S_n\), using the formula:
\[S_n = \frac{n}{2} \begin{pmatrix} u_1 + u_n \end{pmatrix}\]
Where \(u_1\) is the first term of the sequence and \(u_n\) is the \(n^{\text{th}}\) term.
So, for example, \(S_{10}\) refers to the sum of the first \(10\) terms, \(S_{250}\) refers to the sum of the first \(250\) terms, ... .
The method for using this formula is illustrated in the tutorial below.
Tutorial
In the following tutorial we review the method for findng the formula for the \(n^{\text{th}}\) term of a linear sequence. Watch it now.
Exercise
- Calculate the sum of the first \(20\) terms of the arithmetic sequence whose formula for the \(n^{\text{th}}\) term is: \[u_n = 1 + (n-1)\times 4\]
- Calculate the sum of the first \(30\) terms of the arithmetic sequence defined by the formula: \[u_n = 4+(n-1)\times (-3) \]
- Calculate the sum of the first \(25\) terms of the arithmetic sequence defined by the formula: \[u_n = 5+ (n-1)\times 7 \]
-
Given an arithmetic sequence defined by the formula:
\[u_n = 11 + (n-1)\times 6\]
Calculate the sum of the \(6^{th}\) to the \(20^{\text{th}}\) terms included.
Hint: this equals to the difference of two sums.
Formula 2
Given an arithmetic sequence, we can calculate the sum of its first \(n\) terms, \(S_n\), using the formula:
\[S_n = \frac{n}{2} \begin{pmatrix} 2.u_1 + (n-1)d \end{pmatrix}\]
Where \(u_1\) is the first term of the sequence and \(d\) is its common difference.
Note: exam questions frequently involve this second formula.
Tutorial
Exercise 2
- Given the sequence whose first few terms are: \[-2,3,8,13,18, \dots \] calculate the sum of its first \(20\) terms.
- Given the sequence whose first few terms are: \[7,2,-3,-8,-13, \dots \] calculate the sum of its first \(100\) terms.
- Given the sequence whose first few terms are: \[1,4,7,10,13, \dots \] calculate the sum of its first \(50\) terms.
- Given the sequence whose first few terms are: \[0,-3,-6,-9,-12, \dots \] calculate the sum of its first \(200\) terms.
Must Know Exercises
We now learn how to solve some tyical exam-type questions involving the sum of the first \(n\) terms of an arithmetic sequence.
Exercise
-
An arithmetic sequence has eigth term equal to \(33\) and the sum of its first fifteen terms is \(660\).
Find the values of the sequence's first term, \(u_1\), and of its common difference, \(d\). -
The sum of the first \(20\) terms of an arithmetic sequence is \(550\).
Given it has first term equal to \(-2\), find the value of the common difference \(d\).
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