Arithmetic Series

A series is a sequence where the goal is to add all the terms together. We will study arithmetic series and geometric series.

Recall: Notation from Sequences:

$a$ is first term

$d$ is difference, the amount we add each time

$n$ is the number of terms in the series

We will also introduce $l$ , which is the last term of the series. Since there are $n$ terms in the series and we have an $n$ th term equation for arithmetic sequences, we have a formula for $l$ :

$l=a+(n-1)d$

Make sure you are happy with the following topics before continuing.

Sequences

Product

A Level Maths Predicted Papers 2025

117

From: £19.99

The MME A level maths predicted papers are an excellent way to practise, using authentic exam style questions that are unique to our papers. Our examiners have studied A level maths past papers to develop predicted A level maths exam questions in an authentic exam format. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country.

View Product

A Level

A-A* A Level Maths Practice Papers

5

£19.99

The MME A-A* A level maths practice papers are excellent for those top achieving students to practise for their exams, using authentic exam style questions that are unique to our practice papers. Our content experts have studied A level maths past papers and specifications to develop specific A-A* A level maths exam questions in an authentic exam style. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country.

View Product

Adding the Terms

The sum of an arithmetic series with $n$ terms is:

$S_{n}=dfrac{n(a+l)}{2}$

We can prove this result:

$begin{aligned}S_{n}&=a+(a+d)+(a+2d)+…+(l-d)+l[1.2em]&=l+(l-d)+(l-2d)+…+(a+d)+a[1.2em]2S_{n}&=(a+l)+(a+l)+(a+l)+…+(a+l)+(a+l)[1.2em]S_{n}&=dfrac{n(a+l)}{2}end{aligned}$

If we substitute in our formula for $l$ , we get:

$S_{n}=dfrac{n}{2}(2a+(n-1)d)$

Sum Notation

$sum$ means sum, and we can use it instead of writing $S_{n}$ to represent arithmetic series.

Example: $sum_{n=1}^{20}(3n+4)$ means the sum up to the $20$ th term of the arithmetic progression defined by $3n+4$ .

Natural Number Arithmetic Progressions

The sum of the first $n$ natural numbers (positive whole numbers) is:

$S_n = 1 + 2 + 3 + … + (n-1) + n$

So $a=1$ , $l = n$ and $n = n$ .

If we put these values into the formula we have already seen, we would get:

$S_n = dfrac{1}{2} n (n+1)$

Example 1: Arithmetic Series in Practice

Jon is training for a marathon. Last week his furthest run was $4$ miles. He plans to increase the length he runs by $1.1$ miles per day. What is the total distance he has run by the time he reaches his goal of $26$ miles?

[2 marks]

$a=4$

$d=1.1$

$l=26$

Find number of terms between $a$ and $l$

$n=dfrac{26-4}{1.1}=dfrac{22}{1.1}=20$

Need to add $1$ because both the first and last terms are included.

$n=21$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{n(a+l)}{2} [1.2em] S_{21}&=dfrac{21(4+26)}{2}[1.2em]&= dfrac{21times 30}{2}[1.2em]&=dfrac{630}{2}[1.2em]&=315 text{ miles} end{aligned}$

Example 2: Sum Notation

Find $sum_{n=1}^{50}(3n+4)$

[2 marks]

This means the sum of the first $50$ terms of the sequence defined by $3n+4$ .

$a=3times 1+4=3+4=7$

$d=3$

$n=50$

Substitute into formula:

$begin{aligned}sum_{n=1}^{50}(3n+4)&=dfrac{50}{2}(2times 7+(50-1)3)[1.2em]&=25(14+49times 3)[1.2em]&=25(14+147)[1.2em]&=25times 161[1.2em]&=4025end{aligned}$

Arithmetic Series Example Questions

Question 1: Consider the sum of the first $k$ natural numbers.

i) Express this in sum notation.

ii) Find a formula for the sum in terms of $k$ .

iii) What is the value of this sum when $k=100$ ?

iv) If the sum has value $55$ , what is the value of $k$ ?

[8 marks]

i) $sum_{n=1}^{k}n$

ii) Arithmetic progression with:

$a=1$

$d=1$

$n=k$

Sub into formula:

$begin{aligned}sum_{n=1}^{k}n&=dfrac{k}{2}(2times 1+(k-1)1)[1.2em]&=dfrac{k}{2}(2+k-1)[1.2em]&=dfrac{k(k+1)}{2}end{aligned}$

iii) $k=100$

$begin{aligned}sum_{n=1}^{100}n&=dfrac{100(100+1)}{2}[1.2em]&=dfrac{100times 101}{2}[1.2em]&=dfrac{10100}{2}[1.2em]&=5050end{aligned}$

iv) $sum_{n=1}^{k}n=55$

$dfrac{k(k+1)}{2}=55$

$k(k+1)=110$

$k^{2}+k=110$

$k^{2}+k-110=0$

$(k-10)(k+11)=0$

$k=10$ or $k=-11$

$-11$ not feasible

$k=10$

Save your answers with

Gold Standard Education

Sign Up Now

Question 2: Find the sum of the first ten terms of the arithmetic series that begins

$3,8,13,18…$

[3 marks]

$a=3$

$d=5$

$n=10$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{10}{2}(2times 3+(10-1)5)[1.2em]&=5(6+9times 5)[1.2em]&=5(6+45)[1.2em]&=5times 51[1.2em]&=255end{aligned}$

Save your answers with

Gold Standard Education

Sign Up Now

Question 3: At the start of every month, Jenny deposits an amount of money into her savings account. The first month she put in $£10$ , the second month $£20$ , the third month $£30$ , and so on. How much does she have saved after $2$ years?

[4 marks]

$2$ years is $24$ months.

$a=10$

$d=10$

$n=24$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{24}{2}(2times 10+(24-1)10)[1.2em]&=12(20+23times 10)[1.2em]&=12(20+230)[1.2em]&=12times 250[1.2em]&=3000end{aligned}$

Save your answers with

Gold Standard Education

Sign Up Now

Question 4: For an arithmetic series with first term $9$ and difference $4$ , the sum is $184$ . Find the number of terms.

[3 marks]

$S_{n}=184$

$a=9$

$d=4$

Substitute into formula:

$dfrac{n}{2}(2times 9+(n-1)4)=184$

$n(18+4n-4)=368$

$n(14+4n)=368$

$4n^{2}+14n=368$

$4n^{2}+14n-368=0$

$2n^{2}+7n-184=0$

$(n-8)(2n+23)=0$

$n=8$ or $n=-dfrac{23}{2}$

$n=-dfrac{23}{2}$ not feasible

$n=8$

Save your answers with

Gold Standard Education

Sign Up Now

Question 5: The sum of the first $k$ natural numbers is $325$ . Find the value of $k$ .

[2 marks]

$dfrac{1}{2}k (k+1) = 325$

$k^2 + k = 650$

$k^2 + k – 650 = 0$

$(k+26)(k-25) = 0$

$k$ cannot be negative so $k = 25$ .

Save your answers with

Gold Standard Education

Sign Up Now

Additional Resources

Exam Tips Cheat Sheet

A Level

Open

Formula Booklet

A Level

Open

Revise

Arithmetic Series

Arithmetic Series

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Adding the Terms

Sum Notation

Natural Number Arithmetic Progressions

Example 1: Arithmetic Series in Practice

Example 2: Sum Notation

Arithmetic Series Example Questions

Additional Resources

Exam Tips Cheat Sheet

Formula Booklet

Specification Points Covered

Related Topics

Sequences

A Level

GCSE

Key Stage 3

Key Stage 1 & 2

Functional Skills

Past Papers

A Level

GCSE

Other

Courses & Exams

By Type

Special Offers

For Teachers

Functional Skills

GCSE Exams

Tuition

A Level

GCSE

All

Revise

Arithmetic Series

Arithmetic Series

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Adding the Terms

Sum Notation

Natural Number Arithmetic Progressions

Example 1: Arithmetic Series in Practice

Example 2: Sum Notation

Arithmetic Series Example Questions

Additional Resources

Exam Tips Cheat Sheet

Formula Booklet

Specification Points Covered

Related Topics

Sequences