In this explainer, we will learn how to calculate the sum of the terms in an arithmetic sequence with a definite number of terms.
Sequences and series are commonly found in nature and can be used to model the spread of a virus or the decline of a population (these two things are not necessarily linked of course!). In the study of pure mathematics, we interest ourselves in finding the general, or th, term of such sequences, as well as the sum of a given number of terms.
We begin by recalling what we mean by an arithmetic sequence.
Definition: Arithmetic Sequences and Series
An arithmetic sequence is a sequence that has a common difference between consecutive terms. The general term, , of an arithmetic sequence with first term and common difference is given by
An arithmetic series is the sum of a given number of terms of an arithmetic sequence.
For instance, the sequence is an example of an arithmetic sequence.
It has a first term and a common difference of 3.
By using the general term with and , the term of this sequence is given by
The corresponding arithmetic series would be
Letβs look at a practical example before deriving a formula for the sum of a given number of terms in an arithmetic series.
Example 1: Finding the Sum of an Arithmetic Series given the First Three Terms
Find the sum of the first 17 terms of the arithmetic series .
Answer
This is an arithmetic series with first term 12. The common difference, , is found by subtracting a term by the term that precedes it:
The general term, , of an arithmetic sequence with first term and common difference is given by
This means the general term of our series is
The final two terms in the partial sum of this series are found by letting and .
When ,
When ,
This means the sum of the first 17 terms, , can be written as
Of course, we could reverse the series and still get the same result:
Notice how each of the 17 numbers in the list can be paired up with a number in the other list to give a sum of 168.
Adding these equations gives
The sum of the first 17 terms can now be found by dividing 2βββ856 by 2:
The sum of the first 17 terms of this arithmetic series is 1βββ428.
This method for finding the sum of a finite arithmetic series can be generalized to an arithmetic series with first term and common difference .
Example 2: Writing an Expression for the Sum of an Arithmetic Sequence
Find an expression for the sum of an arithmetic sequence whose first term is and whose common difference is .
Answer
We recall that the general term, , of an arithmetic sequence with first term and common difference is given by
We can use this formula to work out the first terms in this sequence.
When ,
When ,
When ,
The pattern continues in this manner.
The sum of the first terms, , is now given as
Of course, if we reverse the series, we will still get the same overall sum:
Notice how each number in the list can be paired up with a number in the
other list to give a constant sum:
This means that when we add the two equations, we will have lots of this expression:
To find an expression for , we then divide through by 2:
Definition: The Sum of an Arithmetic Sequence
The sum of the first terms of an arithmetic series whose first term is and whose common difference is is given by , where
There will be occasions when we are given the first and last terms of an arithmetic series and asked to calculate its sum. We will now consider how to derive a formula for this sum using the formula for the sum of the first terms.
Example 3: Writing an Expression for the Sum of the First π Terms of an Arithmetic Sequence
Write an expression for the sum of the first terms of an arithmetic sequence with first term and last term .
Answer
We begin by recalling the formula that allows us to find the term of an arithmetic series with first term and common difference :
We also know that the sum of the first terms of an arithmetic series whose first term is and whose common difference is is given by , where
By writing as , we can substitute in, as shown:
In a sequence with terms, is the last term. This means we can replace with to find a formula for the sum of the first terms of an arithmetic sequence with first term and last term :
Definition: The Sum of an Arithmetic Sequence
The sum of the first terms of an arithmetic sequence with first term and last term is given by , where
We will now learn how to apply this formula to find the sum of the terms in a finite arithmetic series.
Example 4: Finding the Sum of an Arithmetic Series given the First and Last Terms
Find the sum of the terms of the 11-term arithmetic sequence whose first term is and last term is .
Answer
Recall that the sum of the first terms of an arithmetic sequence with first term and last term is given by , where
The first term in our sequence is and the last term is , so we will let and .
There are 11 terms in the sequence, so we will let .
Then, the sum of the first 11 terms is given by , where
The sum of the terms of this 11-term arithmetic sequence is .
We will now investigate how we can use the general term of an arithmetic sequence to calculate the sum of a given number of terms of the same arithmetic sequence.
Example 5: Finding the Sum of an Arithmetic Sequence given the General Term
Find the sum of the first 10 terms of the sequence , where .
Answer
We are given the general term of the sequence, . This is a formula that allows us to calculate any term given its position number.
For instance, the first term is found by substituting into the formula.
When ,
When ,
When ,
The first three terms of the sequence are 6, 8, and 10. We can therefore deduce the first term to be 6 and the common difference to be 2.
The sum of the first terms of an arithmetic series whose first term is and whose common difference is is given by , where
Since we are calculating the sum of the first 10 terms, we will let , , and :
The sum of the first 10 terms of this sequence is 150.
It is worth noting that the formulae for working with sequences and series can be adapted when a term or a number of terms are given as algebraic expressions. In our next example, we will see what that could look like.
Example 6: Finding the Sum of a Given Arithmetic Sequence in terms of π
Find, in terms of , the sum of the arithmetic sequence .
Answer
In order to find the sum of an arithmetic sequence, we need to know either the number of terms or the value of the last term. In this example, we are given the last term of the arithmetic sequence as an algebraic expression, .
In order to establish the number of the terms in this sequence, letβs look at the general term . When , , which is the first term in the sequence. When , , which is the second term in the sequence. This pattern continues, meaning that, as long as , is also the number of terms in the sequence.
This means we can use the formula to find the sum of the first terms of an arithmetic sequence with first term and last term :
Substituting and into our formula,
The sum of the arithmetic sequence is .
In our final example, we look at how we find the sum of a given number of terms of an arithmetic sequence given information about its terms. This process will involve some problem solving to βwork backwardβ to a solution.
Example 7: Finding the Sum of a Given Number of Terms of an Arithmetic Sequence under a Given Condition
Find the sum of the first 21 terms of an arithmetic sequence given and .
Answer
There are two formulae that we can use to find the sum of a given number of terms of an arithmetic sequence. The first requires us to know the value of the first term, , and the common difference, : while the second requires us to know the value of the first term, , and the last term, :
We have been given information about three of the terms in the sequence, so it follows that we might need to apply the formula for the term of an arithmetic sequence. Since this formula uses the value of the first term and the common difference, we might deduce that we will need the first version of the summation formula.
The general term, , of an arithmetic sequence with first term and common difference is given by
Letting ,
For ,
And for ,
Letβs substitute each of these expressions into the two equations given to us in the question.
The first gives us
Then, the equation becomes
Notice that we have a pair of linear simultaneous equations. These can be solved by subtracting one from the other:
Finally, we can substitute into either of our original equations:
The sum of the first terms of an arithmetic series whose first term is and whose common difference is is given by , where
Since we are finding the sum of the first 21 terms, we will substitute , , and into this formula:
Key Points
- The sum of the first terms of an arithmetic series whose first term is and whose common difference is is given by , where
- The sum of the first terms of an arithmetic sequence with first term and last term is given by , where
