Arithmetic Sequences

  2. Algebra
  3. Arithmetic Sequences
    4. 30'

  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.

Key Concepts

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
 

Essentials

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.

Notes from the video

 

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?

Notes from the video

 

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?

Notes from the video

 

Arithmetic Series

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.

Notes from the video

 

How many Terms?

In the following video, we look at a typical exam-style question which involves the sum of an arithmetic series.

For the following sequence \(28 , 25 , 22 ,...\)

  1. Which is the first negative term of the sequence?
  2. Which is the first term of the series that makes the sum of the series become negative?

Notes from the video

 
 

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

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Exam-style Questions

Question 1

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

Full Solution

 

Question 2

Three consecutive terms of an arithmetic sequence are \(x-3 \ , \ 12 \ ,\ 3x-5\)

Find \(x\)

Hint

Full Solution

 

 

Question 3

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 terms of this sequence exceeds 1000. Find the least value of n

Hint

Full Solution

 

Question 4

Find the sum of all the integers between 100 and 1000 that are divisible by 9

Hint

Full Solution

 

Question 5

An arithmetic sequence has first term U1 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 Un.

c) Find the value of \(\sum _{ r=4 }^{ 17 }{ { U }_{ n } } \)

Hint

Full Solution

 

Question 6

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

Full Solution

 

Question 7

The first terms of a sequence are log3 x , log3 x2 , log3 x3 , ...

Find x if the sum of the first 9 terms is 135

Hint

Full Solution

 
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