adding a common difference . The sequence in the image 1, 5 , 9 has a common difference of 4 since we add 4 to the previous term. There are formula in the booklet to help you with this topic, but it is better not to be too reliant on them. Exam questions often use other areas of the mathematics course. We will get practice in all of these ideas on this page.
On this page, you should learn to
Use the formula for the nth term of an arithmetic sequence Use the formula for the nth term of an arithmetic series Use \(\sum\) notation for sums The following videos will help you understand all the concepts from this page
Arithmetic Sequence - nth Term
In the following video, we will see where the formula for the nth term of an arithmetic sequence comes from.
Arithmetic Sequences - Finding Terms
In the following video, we look at a typical exam-style question about arithmetic sequences. We do not have to be too reliant on the formula for the nth term to be able to find terms in a sequence.
The 5th term of an arithmetic sequence is 35 and the 10th term is 85.
What is the 20th term?
Arithmetic Sequence - Problem Solving
In the following video, we will look at a typical exam-style question. We can use the information in the question to set up and solve two equations. We are going to concentrate on how we can make the equations easier to solve.
Three numbers are consecutive terms in an arithmetic sequence.
They add to give 45 and when they are multiplied together we get 2640.
What are the three numbers?
When we add terms of a sequence together, we call it a series:
arithmetic sequence 1 , 4 , 7 , 10
arithmetic series 1 + 4 + 7 + 10
There are two useful formulae for the sum to n terms of a arithmetic series
\({ S }_{ n }=\frac { n }{ 2 } \left( { U }_{ 1 }+{ U }_{ n } \right) \)
\({ S }_{ n }=\frac { n }{ 2 } \left( { 2U }_{ 1 }+(n-1)d \right) \)
In the following video, we look at a typical exam-style question which involves the sum of an arithmetic series.
The sum of the first 11 terms of an arithmetic series is 3 times the sum of the first 5 terms.
The 8th term is 53. Find the common difference.
In the following video, we look at a typical exam-style question which involves the sum of an arithmetic series.
For the following sequence \(28 , 25 , 22 ,...\)
Which is the first negative term of the sequence? Which is the first term of the series that makes the sum of the series become negative?
Here is a quiz that practises the skills from this page
The following is an arithmetic sequence. Find a b
a b
2 20
In an arithmetic sequence, U3 = 18 and U6 = 6
Find U1
-2 , 1 , 4 , ...
The nth term, Un = a d
Un = U1 + (n - 1)d
U1 = first term
d = common difference
U1 and U5 are the first and fifth terms of an arithmetic sequence
U1 =11
U5 = 39
U10 = a d
U2 and U7 are terms of an arithmetic sequence
U2 = - 2
U7 = 23
23 - (-2)= 5d
25 = 5d
5 = d
U11 = U7 + 4d
U11 = 23 + 20
U11 = 43
32 , 29 , 26 , ...
The nth term of this sequence is -4
Find n
d = -3
-4 - 32 = -36
-36 = 12(-3)
There are 12 differences. It is the 13th term
An arithmetic series has a first term of 8 and a common difference of 4.
What is the sum of the first 12 terms?
S12 = a b c
Find a b c
\(S_{ n }=\frac { n }{ 2 } (2U_{ 1 }+(n−1)d)\)
\(S_{ 12}=\frac { 12 }{ 2 } (2 \times 8+(12−1)4)\)
The first and second terms of an arithmetic sequence are 4 and 9. The nth term is 79.
d = 5
79 - 4 = 75
75 = 15
There are 15
\(S_{ n }=\frac { n }{ 2 } (U_{ 1 }+U_{ n})\)
\(S_{ 16 }=\frac { 16 }{ 2 } (4+79)= 664\)
Find the sum of the following arithmetic series
(-5) + (-1) + ... + 71
d = 4
Find the number of terms in the sequence:
71 - (-5) = 76
76 = 19 x4
There are 19 differences, so 71 is the 20th term. n = 20
\(S_{ n }=\frac { n }{ 2 } (U_{ 1 }+U_{ n})\)
\(S_{ 20 }=\frac { 20 }{ 2 } (-5+71)= 660\)
Sn is the sum of the first n terms of an arithmetic series
S1 = 1 , S2 = 5
Find S4
S1 = 1 , S2 = 5
U1 = 1 , U2 = 4
Sequence is 1 , 4 , 7 , 10
Sum = 1 + 4 + 7 + 10 = 22
In an arithmetic sequence, the first term is 4 and the third term is 16.
a) Find the common difference
b) Find the 8th term
c) Find the sum of the first 8 terms
Hint a) How many differences are there between the first and the third term?
b) Un = U1 + (n-1)d
c) \({ S }_{ n }=\frac { n }{ 2 } \left( { U }_{ 1 }+{ U }_{ n} \right) \)
Full Solution
Three consecutive terms of an arithmetic sequence are \(x-3 \ , \ 12 \ ,\ 3x-5\)
Find \(x\)
Hint Full Solution
The 2nd term of an arithmetic sequence is 19 and the 5th term is 37.
a) Find the 10th term
b) The sum of the first n n
Hint a) Find d
b) You can solve this algebraically or you could make good use of the table function in the your graphical calculator.
Full Solution
Find the sum of all the integers between 100 and 1000 that are divisible by 9
Hint The challenge is to recognise that this is a question that requires finding the sum of an arithmetic series.
What is the first integer after 100 that is divisible by 9?
What is the last integer before 1000 that is divisible by 9?
Full Solution
An arithmetic sequence has first term U1 d
a) Show that \(U_1+8d=8\)
The sum of the 2nd and the 3rd terms is 42.
b) Find d
The nth term of the sequence is Un
c) Find the value of \(\sum _{ r=4 }^{ 17 }{ { U }_{ n } } \)
Hint a) \(S_{ n }=\frac { n }{ 2 } (2U_{ 1 }+(n−1)d)\)
b) \(U_{ 2 }+U_{ 3 }\ =\ (U_{ 1 }+d)+(U_{ 1 }+2d)\)
c) sum of the first 17 terms - sum of the first 3 terms
Full Solution
In an arithmetic sequence, the 9th term is 4 times the 5th term. The sum of the first 2 terms is -13.
Find the 10th term
Hint Set up and solve two simultaneous equations using the information given:
U9 = 4U5 and S2 = -13
There is no need to use the full formula for the information about the sum since S2 = U1 + U2
Full Solution
The first terms of a sequence are log3 x , log3 x2 , log3 x3 , ...
Find x
Hint The terms in this sequence are arithmetic . This question requires good knowledge of the laws of logarithms logc a + logc b = logc ab
logc a - logc b = logc \(\frac{a}{b}\)
You should be able to find an expression fo rthe common difference and then use this Full Solution MY PROGRESS
Self-assessment How much of Arithmetic Sequences have you understood?
My notes
Which of the following best describes your feedback?
These are sequences where you go from term to term by adding a common difference. The sequence in the image 1, 5 , 9 has a common difference of 4 since we add 4 to the previous term. There are formula in the booklet to help you with this topic, but it is better not to be too reliant on them. Exam questions often use other areas of the mathematics course. We will get practice in all of these ideas on this page.
Twitter 
Facebook 
LinkedIn