Websink Prep UKKS3 MathsYear 7 chaptersMaths textbooksReceptionYear 1Year 2Year 4Year 5Year 6Year 3Year 7Year 8Year 9GCSEBrowse allPast papersFind a tutorBecome a tutor
KS3 Maths · Year 7Chapter 7 · Sequences and graphs
Year 7 · Chapter 7

Sequences and graphs

By the end of this chapter you will be able to

  • Plot and read points in all four quadrants, and say which quadrant a point is in
  • Continue a sequence from a term-to-term rule, and find a term from a position-to-term rule
  • Find the nth term of an arithmetic sequence, and check whether a number is in it
  • Recognise geometric sequences and other sequences, such as square and triangular numbers
  • Find the gradient and y-intercept of a straight line, and write its equation from two points
  • Read a value or a meeting point from a graph, and recognise the shape of a simple quadratic or reciprocal curve

You should already know

  • Negative numbers on a number line (Chapter 1)
  • Substitution and rearranging formulae (Chapters 4 and 6)
  • Coordinates in the first quadrant (Year 5, Chapter 7)

Where this leads

  • Quadratic graphs and their roots, and simultaneous equations (Years 8 and 9)
  • Gradient as a rate of change in context (Chapter 8)

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: A8 coordinates in all four quadrants, A9 recognise, sketch and produce graphs of linear functions, and simple quadratic functions, from tables of values, A10 interpret relationships algebraically and graphically, A11 reduce a linear equation in two variables to y = mx + c, and find the gradient and the intercept, A12 use linear graphs to find the point where two lines meet (approximate solutions to simultaneous equations), A13 read approximate values from graphs of simple functions, including reciprocal and exponential forms in context, A14 generate terms from term-to-term and position-to-term rules, A15 recognise arithmetic sequences and find the nth term, A16 recognise geometric and other sequences that arise in practice.

Exercise P  Do you remember?

Non-calculator
1
FWork out 3 − 7.
2
FWork out 2 × (−3).
3
FWhat is the next term of 2, 4, 6, 8, __?
4
FWork out 10 − 2 × 3.
5
FWrite the coordinates of the point (2, 5), where x = 2 and y = 5.
7.1

Coordinates in all four quadrants

Year 7
A8
Video lesson: Coordinates in all four quadrantsComing soon
A treasure map uses a grid with a centre point called the origin. A point at (−3, 2) is 3 squares to the left and 2 squares up. Negative coordinates are needed when the map goes below or to the left of the centre.

A point is written as (x, y): the x-coordinate first, then the y-coordinate. The axes cross at the origin, (0, 0). The axes divide the plane into four quadrants, numbered 1 to 4 anticlockwise, starting at the top right.

−5−4−3−2−112345−5−4−3−2−112345AB

Key ideas

  • (x, y): the x-coordinate is across, and the y-coordinate is up.
  • First quadrant: x and y both positive. Second: x negative, y positive. Third: both negative. Fourth: x positive, y negative.
  • A point on the x-axis has y = 0; a point on the y-axis has x = 0.
  • Read a point by moving across first, then up or down.
Worked example 7.1.1
Which quadrant is the point (−3, 2) in? (Point A above.)

Method. Look at the signs of the coordinates.

  1. Look at x. x = −3 is negative, so the point is to the left.
    x negative
  2. Look at y. y = 2 is positive, so the point is above the x-axis.
    y positive
  3. Write the answer. Check it against the question.
    The second quadrant

Check. Top left is quadrant 2. ✓

Watch out. Quadrant 1 is top right, and the numbers go anticlockwise.

Worked example 7.1.2
Write the coordinates of point B on the grid.

Method. Read across, then up or down.

  1. Read across. B is 4 units to the right of the origin.
    x = 4
  2. Read up or down. B is 1 unit below the x-axis.
    y = −1
  3. Write the answer. Check it against the question.
    (4, −1)

Check. 4 right and 1 down: (4, −1). ✓

Watch out. The y-coordinate is negative below the x-axis.

Worked example 7.1.3
Plot the point (−4, −1). Which quadrant is it in?

Method. Look at the signs of the coordinates.

  1. Look at x. x = −4 is negative, so the point is to the left.
    x negative
  2. Look at y. y = −1 is negative, so the point is below the x-axis.
    y negative
  3. Write the answer. Check it against the question.
    The third quadrant

Check. Bottom left is quadrant 3. ✓

Watch out. Both negative means bottom left.

Watch out

Watch out: (−2, −3) is in the third quadrant, not the first. Both signs are negative there.

Exam tip

Grown-up tip. Ask the pupil to put a finger on the origin, then walk across and up, saying "across, up" for each point.

Exercise 7A  Coordinates and quadrants

Non-calculator
1
FA point is at (−3, 2). Which quadrant is it in?
2
FWhich quadrant is the point (−4, −1) in?
3
FWhich quadrant is the point (5, −2) in?
4
FWrite the coordinates of point A on the grid shown.
−5−4−3−2−112345−5−4−3−2−112345A
5
PSThe points (−2, 1), (3, 1) and (3, −2) are the corners of a triangle. Find its area, in square units.
6
FWhich of these points lies on the x-axis: (2, 0), (0, 3) or (−1, −1)?
7
FWrite the coordinates of the point on the y-axis, 4 units above the origin.
8
RSam says the point (−2, −3) is in the first quadrant. Explain whether Sam is right.
9
FFor each point say which axis it is on: (a) (−5, 0) (b) (0, −7)
(a)(a) (−5, 0)
(b)(b) (0, −7)
10
PSStart at (−1, 4). Move 3 right and 2 down. Write the coordinates of the new point.
11
PSThe corners of a square are (1, 1), (−1, 1), (−1, −1) and (1, −1). What is the side length of the square?
12
RExplain how you know the four points (1, 1), (−1, 1), (−1, −1) and (1, −1) make a square, using their coordinates only.
7.2

Term-to-term and position-to-term rules

Year 7
A14
Video lesson: Term-to-term and position-to-term rulesComing soon
A pattern of tiles starts with 5 tiles, then adds 4 tiles each time. Each new tile count is a term in a sequence. Two kinds of rule describe it: one says how to get the next term from the last one, and the other says how to find any term from its position.

A term-to-term rule says how to get the next term from the one before, such as "add 4". A position-to-term rule gives any term straight from its position n, such as 4n + 1. For position 3, 4n + 1 = 13.

Key ideas

  • Term-to-term: go from one term to the next, for example "add 4" or "multiply by 2".
  • Position-to-term: put the position n into the rule, for example 4n + 1 with n = 3 gives 13.
  • The term number is the position: the 1st term is at position 1.
  • Check a rule by testing the first two or three terms.
Worked example 7.2.1
Find the next two terms of 5, 9, 13, 17, __, __.

Method. Find the rule from two terms, then apply it.

  1. Find the difference. 9 − 5 = 4, and 13 − 9 = 4, so the rule is add 4.
    17 + 4 = 21
  2. Apply the rule twice. Add 4 to get the next terms.
    21 + 4 = 25
  3. Write the answer. Check it against the question.
    21, 25

Check. 21 − 17 = 4 and 25 − 21 = 4. ✓

Watch out. The rule is add 4 each time.

Worked example 7.2.2
A sequence has the rule 3n + 2. Find the 10th term.

Method. Put n = 10 into the rule.

  1. Substitute n = 10. The 10th term is at position 10.
    3 × 10 + 2
  2. Work out. Multiply first, then add.
    30 + 2 = 32
  3. Write the answer. Check it against the question.
    32

Check. Terms 1 to 3 are 5, 8, 11, so the rule adds 3 each time. ✓

Watch out. Multiply before adding.

Worked example 7.2.3
The sequence 7, 12, 17, 22 has the rule "add 5". Write the 5th term.

Method. Keep adding 5.

  1. Add 5 to the 4th term. The 4th term is 22.
    22 + 5
  2. Work out. This is the 5th term.
    27
  3. Write the answer. Check it against the question.
    27

Check. 27 − 22 = 5. ✓

Watch out. Count the terms carefully.

Watch out

Watch out: a term-to-term rule goes from one term to the next, but a position-to-term rule uses the position n. Do not put the term in place of n.

Exam tip

Grown-up tip. Ask the pupil to build the pattern with counters or matches, one step at a time, and to say how many are added each time.

Exercise 7B  Sequences from rules

Non-calculator
1
FContinue the sequence 2, 5, 8, 11, __, __.
2
FContinue the sequence 30, 27, 24, __.
3
FContinue the sequence 1, 4, 9, 16, __, __.
4
FWrite the first five terms of the sequence that starts at 3 and has the rule "add 4".
5
FA sequence has the rule 3n. Write its first four terms.
6
FA sequence has the rule 5n + 2. Find the 20th term.
7
PSA sequence has the rule 4n + 1. Is 41 a term of this sequence? If it is, write its position.
8
PSA sequence has the rule 3n + 2. Is 50 a term of this sequence?
9
F(a) Write the term-to-term rule of 7, 12, 17, 22. (b) Find the 20th term of 5n + 2.
(a)(a) 7, 12, 17, 22
(b)(b) 5n + 2, the 20th term
10
PSA pattern of matchsticks: 1 square uses 4 sticks, 2 squares use 7 sticks, 3 squares use 10 sticks. The rule is 3n + 1. How many sticks are needed for 10 squares?
11
RSam says the 5th terms of 3n and 3n + 1 are the same. Is Sam right? Explain.
7.3

The nth term of an arithmetic sequence

Year 7
A15
Video lesson: The nth term of an arithmetic sequenceComing soon
A car park charges £5 for the first hour and then £3 for each hour after that. The sequence of prices is 5, 8, 11, 14 and so on. To find the price for any hour without listing every term, we need the nth term.

An arithmetic sequence goes up or down by the same amount each time. This amount is the common difference, d. The nth term is dn + (first term − d). For 5, 8, 11, 14: d = 3 and 5 − 3 = 2, so the nth term is 3n + 2.

Key ideas

  • The coefficient of n is the common difference d.
  • The constant is the first term minus d.
  • Check the nth term: put n = 1 and n = 2 and see if the terms match.
  • A number is in the sequence if n = (number minus constant) ÷ d is a whole number.
Worked example 7.3.1
Find the nth term of 5, 8, 11, 14.

Method. Find the common difference first. The nth term is dn + (first term − d), and check it with the terms given.

  1. Find the difference between terms. Each term goes up or down by the same amount, 3.
    8 − 5 = 3, 11 − 8 = 3
  2. Write the coefficient of n. The coefficient of n is the common difference.
    d = 3, so the answer starts with 3n
  3. Find the constant part. The first term is 5, and 5 − 3 = 2.
    5 − 3 = 2
  4. Write the nth term. Put the parts together.
    3n + 2

Check. Check: with n = 1, 3n + 2 gives 5. ✓

Watch out. The constant part is the first term minus the difference, not the first term itself.

Worked example 7.3.2
Find the nth term of 20, 17, 14, 11.

Method. Find the common difference first. The nth term is dn + (first term − d), and check it with the terms given.

  1. Find the difference between terms. Each term goes up or down by the same amount, −3.
    17 − 20 = −3, 14 − 17 = −3
  2. Write the coefficient of n. The coefficient of n is the common difference.
    d = −3, so the answer starts with −3n
  3. Find the constant part. The first term is 20, and 20 − −3 = 23.
    20 − −3 = 23
  4. Write the nth term. Put the parts together.
    −3n + 23

Check. Check: with n = 1, −3n + 23 gives 20. ✓

Watch out. The constant part is the first term minus the difference, not the first term itself.

Worked example 7.3.3
Is 100 a term of the sequence 5n − 4? Explain.

Method. Set 5n − 4 equal to 100 and solve for n.

  1. Set up the equation. Find n with 5n − 4 = 100.
    5n = 104
  2. Solve. Divide by 5.
    n = 20.8
  3. Write the answer. Check it against the question.
    No, because n = 20.8 is not a whole number

Check. 5 × 20 − 4 = 96 and 5 × 21 − 4 = 101, so 100 is between terms. ✓

Watch out. n must be a whole number.

Watch out

Watch out: for 20, 17, 14 the difference is −3, so the coefficient is −3, and the constant is 20 + 3 = 23: the nth term is −3n + 23.

Exam tip

Grown-up tip. Ask the pupil to write the difference between the first two terms, and to say it out loud before finding the nth term.

Exercise 7C  The nth term of a sequence

Non-calculator
1
FFind the nth term of 3, 7, 11, 15.
2
FFind the nth term of 5, 9, 13.
3
FFind the nth term of 2, 6, 10, 14.
4
FFind the nth term of 10, 12, 14, 16.
5
FFind the nth term of 1, 6, 11, 16.
6
FFind the nth term of 20, 17, 14, 11.
7
FFind the nth term of 7, 4, 1, −2.
8
FFind the 25th term of 4n + 3.
9
PSIs 52 a term of the sequence 4n + 3? Explain.
10
FA sequence is 6, 11, 16, 21, and so on. (a) Find the nth term. (b) Which term is 46?
(a)(a) nth term of 6, 11, 16, 21
(b)(b) which term is 46
11
PSThe nth term of a sequence is 2n + 9. Which term is 35?
12
RExplain why the sequence 3, 6, 12, 24 cannot be written as dn + c.
7.4

Geometric and other sequences

Year 7
A16
Video lesson: Geometric and other sequencesComing soon
A colony of bacteria doubles every hour. It starts with 5 bacteria, so the counts are 5, 10, 20, 40 and so on. Sequences like this are not made by adding the same amount each time.

A geometric sequence has a common ratio: each term is multiplied by the same number, such as 2, 3 or ½. Other sequences follow different patterns: square numbers 1, 4, 9, 16; cube numbers 1, 8, 27; triangular numbers 1, 3, 6, 10, where each term adds one more than the last; and the Fibonacci pattern, where each term is the sum of the two before it.

Key ideas

  • Geometric: multiply by the same number each time (the common ratio).
  • Square numbers: 1, 4, 9, 16, 25. Cube numbers: 1, 8, 27, 64.
  • Triangular numbers: 1, 3, 6, 10, 15, 21.
  • Fibonacci-type: add the two terms before to get the next term: 1, 1, 2, 3, 5, 8.
Worked example 7.4.1
Continue the sequence 2, 6, 18, 54, __.

Method. Look at what is happening from term to term.

  1. Find the ratio. The terms are multiplied by 3 each time: 6 ÷ 2 = 3.
    2 × 3 = 6, 6 × 3 = 18
  2. Apply it. Multiply 54 by 3.
    54 × 3 = 162
  3. Write the answer. Check it against the question.
    162

Check. 162 ÷ 54 = 3. ✓

Watch out. Multiplying by 3 is not adding 3.

Worked example 7.4.2
Write the first five triangular numbers.

Method. Each term adds one more than the last.

  1. Start with 1. The first triangular number is 1.
    1
  2. Add 2, then 3, then 4, then 5. The amount added goes up by one each time.
    1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15
  3. Write the answer. Check it against the question.
    1, 3, 6, 10, 15

Check. Adding 5 to 10 gives 15. ✓

Watch out. The amount added increases by 1 each time.

Worked example 7.4.3
A sequence starts 1, 1, 2, 3, 5. What are the next two terms?

Method. Each term is the sum of the two before it.

  1. Add the last two terms. 3 + 5 is the next term.
    5 + 3 = 8
  2. Add the next two. Now add the last two terms again.
    8 + 5 = 13
  3. Write the answer. Check it against the question.
    8, 13

Check. 13 is 5 + 8. ✓

Watch out. Add the two numbers before, not the number before and the number after.

Watch out

Watch out: 2, 4, 8, 16 is geometric (multiply by 2). It is not arithmetic, so the difference is not the same each time.

Exam tip

Grown-up tip. Ask the pupil to find the ratio by dividing each term by the one before, and to compare with the difference.

Exercise 7D  Geometric and other sequences

Non-calculator
1
FContinue the sequence 3, 6, 12, 24, __.
2
FContinue the sequence 5, 15, 45, __.
3
FContinue the sequence 81, 27, 9, __.
4
FContinue the sequence 2, −4, 8, −16, __.
5
FContinue the sequence 1, 1, 2, 3, 5, 8, __, __.
6
FWrite the next triangular number after 21 in the sequence 1, 3, 6, 10, 15, 21, __.
7
FWrite the first five square numbers.
8
FWrite the first four cube numbers.
9
FA bacteria count doubles every hour. It starts at 5. Write the first four terms.
10
FA sequence has the rule: multiply by 3 each time. (a) Write the common ratio. (b) Find the next term after 54.
(a)(a) the common ratio
(b)(b) the next term after 54
11
PSA sequence starts 4, 12, 36. Find the 5th term.
12
RIs 16 a term of the sequence 2, 4, 8, __? Explain.
7.5

Straight-line graphs

Year 7
A9, A11
Video lesson: Straight-line graphsComing soon
A taxi firm charges £3 plus £2 for each mile. The total cost C pounds for m miles is C = 2m + 3. A graph of this rule is a straight line, and its steepness and starting height can be read from the equation.

A linear equation in x and y is a straight line when it is written as y = mx + c. The number m is the gradient, which measures how steep the line is: it is how much y goes up for each 1 step across. The number c is the y-intercept, where the line crosses the y-axis.

−11234−112345678910

Key ideas

  • y = mx + c: m is the gradient (steepness), c is the y-intercept (where the line crosses the y-axis).
  • Gradient = change in y ÷ change in x, between any two points on the line.
  • To draw a line, make a table of values, plot the points, and join them with a straight line.
  • Parallel lines have the same gradient.
Worked example 7.5.1
Find the gradient and y-intercept of y = 3x − 1.

Method. Compare with y = mx + c.

  1. Compare. The number in front of x is m, and the number on its own is c.
    m = 3, c = −1
  2. Read the intercept. The line crosses the y-axis at (0, −1).
    y-intercept −1
  3. Write the answer. Check it against the question.
    gradient 3, y-intercept −1

Check. Put x = 0: y = −1. ✓

Watch out. The intercept keeps its sign.

Worked example 7.5.2
Find the equation of the line through (1, 2) and (3, 8).

Method. Find the gradient, then the intercept.

  1. Find the gradient. Gradient = change in y ÷ change in x = (8 − 2) ÷ (3 − 1).
    6 ÷ 2 = 3
  2. Find the intercept. Use one point: 2 = 3 × 1 + c, so c = −1.
    c = 2 − 3 = −1
  3. Write the equation. The equation is y = mx + c.
    y = 3x − 1
  4. Write the answer. Check it against the question.
    y = 3x − 1

Check. Check: 3 × 3 − 1 = 8. ✓

Watch out. Check that both points fit the equation.

Worked example 7.5.3
Find the gradient of the line through (0, 2) and (4, 10).

Method. Gradient = change in y ÷ change in x.

  1. Find the changes. y goes from 2 to 10 and x goes from 0 to 4.
    10 − 2 = 8, 4 − 0 = 4
  2. Divide. The gradient is 8 ÷ 4.
    2
  3. Write the answer. Check it against the question.
    2

Check. Going 4 across and 8 up: 8 ÷ 4 = 2. ✓

Watch out. Change in y first, divided by change in x.

Watch out

Watch out: y = 3x and y = 3x + 2 are parallel: they have the same gradient but cross the y-axis in different places.

Exam tip

Grown-up tip. Ask the pupil to read the gradient as "up 3 for each 1 across", and to point to the intercept on the y-axis.

Exercise 7E  Straight lines: gradient and intercept

Non-calculator
1
FWrite the gradient and the y-intercept of y = 4x + 1.
2
FWrite the gradient and the y-intercept of y = 3x − 5.
3
FWrite the gradient and the y-intercept of y = −2x + 6.
4
FThe line y = x passes through the origin. Write its gradient and y-intercept.
5
FFind the gradient of the line through (0, 2) and (4, 10).
6
FFind the equation of the line through (1, 2) and (3, 8).
7
FWhich of these lines is parallel to y = 2x + 3? A: y = 2x − 1 B: y = 3x + 2 C: y = x + 3
8
FComplete the table for y = 2x − 3: x = −1, 0, 1, 2 gives y = ?, ?, ?, ?
9
RDoes the point (2, 9) lie on the line y = 4x + 1? Explain.
10
F(a) Write the gradient of y = 5 − 2x. (b) Write the y-intercept of y = 3x.
(a)(a) gradient of y = 5 − 2x
(b)(b) y-intercept of y = 3x
11
PSA taxi charges £3 plus £2 per mile. Write the rule in the form C = mx + c, and say what m and c mean.
12
RSam says the lines y = 3x + 2 and y = 3x − 5 are the same line. Is he right? Explain.
7.6

Reading and solving from graphs

Year 7
A9, A10, A12, A13
Video lesson: Reading and solving from graphsComing soon
Two friends walk towards each other along a path. Their distances from the start are given by two straight-line rules. The point where the two lines cross is when they meet, and its coordinates give the time and place.

When two straight lines cross, the crossing point (x, y) fits both equations. So we can find it by setting the two rules equal, or by reading it from a graph.

The graph of y = x2 is a U-shaped curve called a parabola. It is symmetrical, and it has its lowest point at the origin.

−3−2−1123−1123456789

Key ideas

  • A meeting point fits both equations: set the two rules equal and solve.
  • The y-coordinate is found by putting the x-value into either equation.
  • y = x2 is a U-shaped curve, symmetrical about the y-axis.
  • Read values from a graph by going across to the line, then up or down to the axis.
Worked example 7.6.1
Find where y = 2x + 1 and y = 7 − x meet.

Method. Set the two equations equal and solve for x.

  1. Set equal. The meeting point has the same y-value on both lines.
    2x + 1 = 7 − x
  2. Solve for x. Add x to both sides, then subtract 1, then divide by 3.
    3x = 6, so x = 2
  3. Find y. Put x = 2 into y = 7 − x.
    y = 7 − 2 = 5
  4. Write the answer. Check it against the question.
    (2, 5)

Check. Check: 2 × 2 + 1 = 5 and 7 − 2 = 5. ✓

Watch out. Put the x-value into either equation to find y.

Worked example 7.6.2
Using y = x2, find y when x = 1.5.

Method. Substitute x = 1.5.

  1. Write the rule. y = x2 means x multiplied by itself.
    y = 1.5 × 1.5
  2. Square 1.5. Multiply 1.5 by itself.
    1.5 × 1.5 = 2.25
  3. Write the answer. Check it against the question.
    2.25

Check. On the curve, the point (1.5, 2.25) lies between (1, 1) and (2, 4). ✓

Watch out. Check the value lies between the values either side.

Worked example 7.6.3
Find where y = 3x and y = 8 − x meet.

Method. Set the two rules equal and solve.

  1. Set equal. 3x = 8 − x.
    4x = 8
  2. Solve. Divide by 4.
    x = 2
  3. Find y. Put x = 2 into y = 3x.
    y = 6
  4. Write the answer. Check it against the question.
    (2, 6)

Check. 8 − 2 = 6. ✓

Watch out. Add x to both sides first.

Watch out

Watch out: the parabola y = x2 is not a straight line. Do not join its points with a ruler.

Exam tip

Grown-up tip. Ask the pupil to find the crossing point on the graph, then check it with the two equations.

Exercise 7F  Reading and solving from graphs

Non-calculator
1
FThe lines y = 2x + 1 and y = 7 − x cross at one point. Find it.
2
FFind the point where y = x + 2 meets y = 8 − 2x.
3
FComplete the table for y = x2 when x = −3, −2, −1, 0, 1, 2, 3.
4
FUsing the curve y = x2, what is y when x = 1.5?
5
RDescribe the shape of the graph of y = x2.
6
FComplete the table for y = 1/x when x = 1, 2, 4. (Give y as a decimal.)
7
FComplete the table for y = 2x when x = 0, 1, 2, 3.
8
FThe lines y = 3x and y = 12 − x cross. (a) Find x. (b) Find y.
(a)(a) x
(b)(b) y
9
PSA line passes through (0, 4) and (2, 10). Find its equation.
10
RExplain why the lines y = 2x + 1 and y = 2x − 3 never meet.

Chapter summary

SkillMethod
Four quadrantsRead (x, y) across first then up or down; signs show the quadrant
Term-to-term ruleSay how to get the next term: add, subtract, multiply or divide
nth term (arithmetic)dn + (first term − d); d is the common difference
Geometric sequenceMultiply by the same ratio each time
Straight liney = mx + c: m is the gradient, c is the y-intercept
GradientChange in y ÷ change in x between two points
Meeting pointSet the two rules equal and solve; put x into either rule to find y
Quadratic and reciprocaly = x2 is a U-shaped curve; y = 1/x gets closer to zero

Can you…?

  • I can plot and read points in all four quadrants
  • I can continue sequences from term-to-term and position-to-term rules
  • I can find the nth term of an arithmetic sequence and check if a number is in it
  • I can recognise geometric, square, cube, triangular and Fibonacci-type sequences
  • I can find the gradient and y-intercept of a straight line, and its equation from two points
  • I can find where two lines meet, and read values from a simple graph

Key terms

origin
the point (0, 0) where the axes cross
quadrant
one of the four regions of the grid, numbered 1 to 4 anticlockwise from the top right
term
one number in a sequence
common difference
the amount added to each term to get the next, in an arithmetic sequence
common ratio
the number each term is multiplied by, in a geometric sequence
gradient
how steep a straight line is: the change in y for a change of 1 in x
y-intercept
the point where a line crosses the y-axis
parabola
the U-shaped curve of y = x2

Exercise 7Z  Check-up

Non-calculator
1
Which quadrant is the point (−2, −6) in?
2
Find the nth term of 4, 8, 12, 16.
3
Write the gradient of y = −3x + 2.
4
Continue the sequence 1, 5, 25, 125, __.
5
Find the 10th term of 5n − 2.
6
Find the point where y = x and y = 6 − x meet.
7
Is (4, 13) on the line y = 3x + 1? Explain.
8
Find the nth term of 9, 7, 5, 3.
9
Find the value of y = x2 when x = −3.
10
Find the equation of the line through (0, 5) and (2, 9).

Total 21 marks. Allow about 21 minutes.

Exercise 7X  Practice test questions

Non-calculator
1
FWhich quadrant is the point (−3, 5) in?[1 mark]
2
FThe nth term of a sequence is 3n + 4. Find the 10th term.[1 mark]
3
FFind the nth term of 7, 10, 13, 16.[2 marks]
4
FWrite the gradient and the y-intercept of y = 5x − 2.[2 marks]
5
FContinue the sequence 2, 6, 18, 54. Write the next two terms.[2 marks]
6
FA sequence has the nth term 12 − 3n. Find the 50th term.[2 marks]
7
FThe line y = 2x − 1 is drawn. Find the value of y when x = 2.[2 marks]
8
FFind the equation of the line through (1, 2) and (3, 8).[3 marks]
9
FFind the point where y = 3x and y = 8 − x meet.[3 marks]
10
RIs 100 a term of the sequence 6n + 1? Explain.[3 marks]
← Chapter 6: Solving equations and using formulaeChapter 8: Ratio, proportion and rates of change →

Websink Prep UK is an independent site. It is not affiliated with, endorsed by or sponsored by AQA, Pearson Edexcel, OCR, WJEC, Eduqas, CCEA, SQA, GL Assessment, CEM or the Department for Education. All exam board names and trademarks belong to their owners. Past papers are linked from the exam boards' own websites and are not hosted here.

About Websink Prep UK · Find a tutor · Privacy · Terms · Cookie settings