Arithmetic Sequences
An arithmetic sequence is a sequence in which the same value, called the common difference, is added to each term to obtain the next term. For example, the sequence 1,5,9,\ldots has common difference 4. There are formulae in the formula booklet to help with this topic, but it is useful to understand how they are constructed and when to use them. Exam questions often connect arithmetic sequences and series with other areas of mathematics, and this page provides practice with these ideas.
The following videos will help you understand all the concepts on this page.
In the following video, we will see where the formula for the nth term of an arithmetic sequence comes from.
Notes from the video
Print from here
Arithmetic Sequences
An arithmetic sequence has a common difference
The nth term of an arithmetic sequence is U_n=\color{#4F81BD}{U_1}+(n-1)\color{red}{d}
Arithmetic Series
A series is formed when the terms of a sequence are added: 1+4+7+10.
S_n=\dfrac{n}{2}(U_1+U_n). This is useful when we know U_n
S_n=\dfrac{n}{2}(2U_1+(n-1)d). This is useful when we do not know U_n
Solving Problems with GDC
You can make good use of your graphical calculator to find out how many terms there are in a sequence. The table function is particularly useful:
Here is a quiz that practises the skills from this page.
Arithmetic Sequences Quiz
In an arithmetic sequence, the first term is 4 and the third term is 16.
a) Find the common difference.
b) Find the eighth term.
c) Find the sum of the first eight terms.
Hint
a) How many differences are there between the first and the third term?
c) { S }_{ n }=\dfrac { n }{ 2 } ( { U }_{ 1 }+{ U }_{ n} )
Full Solution
The 2nd term of an arithmetic sequence is 19 and the 5th term is 37.
a) Find the tenth term.
b) The sum of the first n terms of this sequence exceeds 1000. Find the least value of n.
Hint
a) Find d by considering how many common-difference intervals there are between the second and fifth terms.
b) You can solve this algebraically or you could make good use of the table function on your graphical calculator.
Full Solution
\begin{aligned} &\text{Our graphical calculator can help}\\ &\text{Plot } y=3x^2+10x\\ &\text{And } y=1000 \end{aligned}
Therefore, the least value is n=17.
We can also use the table function:
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 U_1 and common difference d. The sum of the first 17 terms is 136.
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 U_n.
c) Find the value of \displaystyle \sum_{r=4}^{17}U_r.
Hint
a) S_n=\dfrac{n}{2}\bigl(2U_1+(n-1)d\bigr)
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 tenth term.
Hint
Set up and solve two simultaneous equations using the information given.
U_9=4U_5 and S_2=-13
There is no need to use the full formula for the information about the sum since S_2=U_1+U_2
Full Solution
The first terms of a sequence are \log_3x,\ \log_3x^2,\ \log_3x^3,\ldots, where x>0.
Find x if the sum of the first nine terms is 135.
Hint
- The terms in this sequence are arithmetic.
- This question requires a good knowledge of the laws of logarithms.
\log_c a+\log_c b=\log_c(ab)
\log_c a-\log_c b=\log_c\bigl(\dfrac{a}{b}\bigr)
- Find an expression for the common difference and then use the arithmetic-series formula.