Introduction
An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed amount, called the common difference.
For example:
Here, the common difference is 3.
The nth term of an arithmetic sequence is given by:
where 
is the first term and d is the common difference.
The sum of the first n terms is:
Example
The 3rd term of an arithmetic sequence is 10, and the 7th term is 22. Find the first term and the common difference.
Solution:
Subtracting the first from the second:
Substitute back:
So the sequence is:
The Linear Nature of Sequences
An arithmetic sequence is essentially a linear function. If you were to plot the terms of a sequence on a graph where the x-axis is the position (
) and the y-axis is the term value (
), the points would form a perfectly straight line.
- The common difference (d) is equivalent to the gradient (slope) of the line.
- If d is greater than 0, the sequence is increasing (sloping upwards).
- If d is less than 0, the sequence is decreasing (sloping downwards).
Recursive vs. Explicit Formulas
Students often confuse these two ways of defining a sequence.
- Recursive Formula: This is the "term-to-term" rule. it tells you how to get to the next number if you already have the current one. It is simple but slow if you need to find the 100th term.
- Explicit Formula: This is the "position-to-term" rule. It allows you to jump directly to any position in the sequence without knowing the numbers in between.
Membership Testing: "Is this number in the sequence?"
A common exam question asks if a specific value (e.g., 105) belongs to a given sequence.
- To solve this, set the value equal to your nth term formula and solve for n.
- The Rule: If n results in a positive integer (a whole number), the value is in the sequence. If n is a fraction or decimal, the value is not part of the sequence because "positions" must be whole numbers.
Practice Questions & Solutions
The fourth term of an arithmetic sequence is 14 and the sixth term is 20. Determine the sequence.
Subtract:
Then:
Sequence:
The first term of an arithmetic sequence is 3 and the fifteenth term is 45. Find the common difference and the sum of the first fifteen terms.
Sum of the first 15 terms:
Common difference:
Sum:
Find the sum of the first fifteen multiples of 7.
Sum:
Find the sum of the first fifteen numbers ending in 5.
Sum:
Find the sum of the first fifteen even numbers greater than 20.
Sum:
Find the angles of a convex quadrilateral, knowing they are in arithmetic sequence and their sum is 360 degrees.
Let the angles be:
If 
and 
, then the angles are:
The shorter leg of a right triangle is 6cm. The sides form an arithmetic sequence. Find the lengths of the other two sides.
Let the sides be:
Given 
:
Solve for d:
Reject the negative value.
Sides:
Find three numbers in an arithmetic sequence whose sum is 15 and the sum of their squares is 83.
Let the numbers be:
Numbers:
An arithmetic sequence is defined by the formula 
. If this sequence were plotted on a coordinate plane, what would be the gradient of the resulting line?
This is in the form:
where m is the common difference. Since the coefficient of n is 5:
The gradient is 5.
Does the number 142 appear in the arithmetic sequence 7, 12, 17, 22...? Show your working.
First, identify the first term and common difference:
Create the n-th term formula:
Now, set the formula equal to 142 and solve for n:
Find the 10th term of the arithmetic sequence: 0.2, 0.5, 0.8, 1.1...
Identify the parameters:
Use the formula for the 10th term:
In an arithmetic progression, the common difference is -4 and the 12th term is 10. Calculate the first term, 
.
Use the general formula:
Substitute the known values:
Rearrange to solve:
A library fine starts at £0.50 on the first day and increases by £0.25 every day after that. Write an explicit formula for the fine on day n and calculate the total fine on day 20.
The formula for the fine on day n is:
To find the fine on day 20, substitute n = 20:
Summarise with AI:
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I get it somehow, there are others who are unfamiliar and I have a hard time with it but it’s okay that I get it.
Thanks for sharing your thoughts! It’s completely normal to find some parts unfamiliar or more difficult than others. The important thing is that you’re making progress and understanding more as you practise. Keep going! 😊
At least I understood some of it, but there are others that I’m confused about and have a hard time understanding, but I get it.
Thanks for sharing! It’s great that you’ve understood some of it already. Don’t worry if a few parts are still confusing — keep practising and take your time with the trickier sections. You’ll get there! 😊
The lesson was a little difficult for me, especially the word problems, but the examples helped me understand it better.
Thanks for your feedback! Word problems can definitely be tricky, so we’re glad the examples helped make the lesson clearer. Keep practising — it gets easier with time! 😊
Thankyouuu po