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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