Algebra: Sequences and Graphs
Sequences and graphs covers finding patterns in number sequences and describing straight-line graphs using coordinates, gradients and y-intercepts. A sequence is an ordered list of numbers following a rule, and its nth term is an expression that gives any term directly from its position number; a straight-line graph has equation y = mx + c, where m is the gradient and c is the y-intercept.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a sequence, find the common (constant) difference between consecutive terms.
- Write the nth term as (common difference) x n, then adjust by adding or subtracting a number so the first term matches.
- To find a term's value, substitute its position number, n, into the nth-term expression.
- To find a position, set the nth-term expression equal to the given value and solve the resulting equation.
- For a straight-line graph y = mx + c, read the gradient, m, as the number multiplying x, and the y-intercept, c, as the constant term.
- To find the equation of a line from a point and gradient, substitute the coordinates into y = mx + c and solve for c.
Worked example
A sequence begins 4, 10, 16, 22, 28, ... Find an expression for the nth term, and use it to find the 20th term.
- Find the common difference: 10 - 4 = 6, 16 - 10 = 6, so the common difference is 6.
- Start with 6n as the basis of the nth term.
- Compare 6n with the sequence: when n = 1, 6n = 6, but the first term is 4, so try 6n - 2 (since 6 - 2 = 4).
- Check: when n = 2, 6(2) - 2 = 12 - 2 = 10, which matches. So the nth term is 6n - 2.
- Substitute n = 20: 6(20) - 2 = 120 - 2 = 118.
Practice questions
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Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The nth term of a sequence is given by 5n - 3. A term in the sequence is 62. Work out its position, n, in the sequence.
Find an expression, in terms of n, for the nth term of the sequence 9, 15, 21, 27, ... Hence determine whether 150 is a term in the sequence, showing your working.
A straight line passes through the points (0, 4) and (2, 10). Find the equation of the line in the form y = mx + c.
Free printable worksheet
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