Sequences and linear graphs

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   9 − 4 = 5
2. Check the other differences. Every difference must be the same for the rule to work.
   9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5
3. Use the rule to find the next term. Add 5 to the last term.
   19 + 5 = 24
4. Write the answer.
   24, 29

Check. The rule is add 5. Each term is 5 more than the one before.

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   44 − 50 = -6
2. Check the other differences. Every difference must be the same for the rule to work.
   44 − 50 = -6, 38 − 44 = -6
3. Use the rule to find the next term. Add -6 to the last term.
   38 + -6 = 32
4. Write the answer.
   32, 26

Check. The rule is add -6. Each term is -6 more than the one before.

Method. Divide each term by the one before it. If the answer is always the same, multiply by that number each time.

1. Divide the second term by the first. This gives the multiplier.
   6 ÷ 3 = 2
2. Check the other terms. The same multiplier must work for each pair.
   6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2
3. Use the rule to find the next term. Multiply the last term by 2.
   24 × 2 = 48
4. Write the answer.
   48, 96

Check. The rule is multiply by 2. Each term is 2 times the term before it.

Method. Divide each term by the one before it. If the answer is always the same, multiply by that number each time.

1. Divide the second term by the first. This gives the multiplier.
   200 ÷ 400 = 1/2
2. Check the other terms. The same multiplier must work for each pair.
   200 ÷ 400 = 1/2, 100 ÷ 200 = 1/2
3. Use the rule to find the next term. Multiply the last term by 1/2.
   100 × 1/2 = 50
4. Write the answer.
   50, 25

Check. The rule is multiply by 1/2. Each term is 1/2 times the term before it.

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   7 − 2 = 5
2. Check the other differences. Every difference must be the same for the rule to work.
   7 − 2 = 5, 12 − 7 = 5, 17 − 12 = 5
3. Use the rule to find the next term. Add 5 to the last term.
   17 + 5 = 22
4. Write the answer.
   22, 27

Check. The rule is add 5. Each term is 5 more than the one before.

Method. Divide each term by the one before it. If the answer is always the same, multiply by that number each time.

1. Divide the second term by the first. This gives the multiplier.
   27 ÷ 81 = 1/3
2. Check the other terms. The same multiplier must work for each pair.
   27 ÷ 81 = 1/3, 9 ÷ 27 = 1/3, 3 ÷ 9 = 1/3
3. Use the rule to find the next term. Multiply the last term by 1/3.
   3 × 1/3 = 1
4. Write the answer.
   1, 1/3

Check. The rule is multiply by 1/3. Each term is 1/3 times the term before it.

Method. Find the rule from the terms you can see, then find the missing term.

1. Find the rule. The difference between the terms that are given is 4.
   4 is the difference
2. Find the missing term. Work out the term that fits the rule.
   3 + 4 = 7
3. Write the answer.
   7

Check. The term after 7 is 11, which matches the sequence.

Method. Find the rule from the terms you can see, then find the missing term.

1. Find the rule. The difference between the terms that are given is -4.
   -4 is the difference
2. Find the missing term. Work out the term that fits the rule.
   20 − 4 = 16
3. Write the answer.
   16

Check. The term after 16 is 12, which matches the sequence.

Method. Find the rule first, then the two missing terms.

1. Find the step. The terms 2 and 8 have two steps between them, so the step is (8 − 2) ÷ 2.
   (8 − 2) ÷ 2 = 3
2. Find the missing terms. Add 3 to each term to get the next one.
   2 + 3 = 5, and 8 + 3 = 11
3. Check. The last term is 11 + 3 = 14.
   14
4. Write the answer.
   5 and 11

Check. 2, 5, 8, 11, 14 all go up by 3.

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   8 − 5 = 3
2. Check the other differences. Every difference must be the same for the rule to work.
   8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3
3. Use the rule to find the next term. Add 3 to the last term.
   14 + 3 = 17
4. Write the answer.
   17, 20

Check. The rule is add 3. Each term is 3 more than the one before.

Method. Find the difference between the terms.

1. Find the difference. Subtract the first term from the second: 25 − 30.
   25 − 30 = −5
2. Check another pair. 20 − 25 = −5, so the difference is the same.
   20 − 25 = −5
3. Write the rule. The sequence goes down by 5 each time.
   Subtract 5

Check. 30 − 5 = 25, and 25 − 5 = 20.

Method. Check the rule on the terms.

1. Check the differences. 4 − 2 = 2, but 8 − 4 = 4.
   2 ≠ 4
2. Look for another rule. 4 ÷ 2 = 2, 8 ÷ 4 = 2, 16 ÷ 8 = 2.
   multiply by 2
3. Write the answer. Ali is not right. The rule is multiply by 2.
   No

Check. 2 + 2 = 4, but the next term is 8, not 6.

Method. Check the rule with the terms.

1. Check multiply by 5. 5 × 5 = 25, which is not 10.
   25 ≠ 10
2. Look for the real rule. Each term is 5 more than the one before.
   add 5
3. Write the answer. Ali is not right. The rule is add 5.
   No

Check. 10 ÷ 5 = 2, which is not the next term 15 ÷ 10 = 1.5.

Method. Check the differences, then the multipliers.

1. Find the differences. 2 − 1 = 1, 4 − 2 = 2, 8 − 4 = 4.
   1, 2, 4
2. Find the multipliers. 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2.
   multiply by 2
3. Write the answer. The differences change, so it is not add a number. The rule is multiply by 2.
   Multiply by 2

Check. 1 × 2 = 2, 2 × 2 = 4, and so on.

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 4
2. Write the first part of the nth term. The nth term starts with the difference times n.
   4n
3. Find the number to add. The first term is 3. Put n = 1 in 4n and find what is left over.
   4 × 1 + ? = 3, so ? = -1 = -1
4. Write the nth term.
   4n − 1

Check. Check n = 2: 4n − 1 gives 7, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 3
2. Write the first part of the nth term. The nth term starts with the difference times n.
   3n
3. Find the number to add. The first term is 5. Put n = 1 in 3n and find what is left over.
   3 × 1 + ? = 5, so ? = 2 = 2
4. Write the nth term.
   3n + 2

Check. Check n = 2: 3n + 2 gives 8, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 4
2. Write the first part of the nth term. The nth term starts with the difference times n.
   4n
3. Find the number to add. The first term is 2. Put n = 1 in 4n and find what is left over.
   4 × 1 + ? = 2, so ? = -2 = -2
4. Write the nth term.
   4n − 2

Check. Check n = 2: 4n − 2 gives 6, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = -3
2. Write the first part of the nth term. The nth term starts with the difference times n.
   -3n
3. Find the number to add. The first term is 10. Put n = 1 in -3n and find what is left over.
   -3 × 1 + ? = 10, so ? = 13 = 13
4. Write the nth term.
   −3n + 13

Check. Check n = 2: −3n + 13 gives 7, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 3
2. Write the first part of the nth term. The nth term starts with the difference times n.
   3n
3. Find the number to add. The first term is 1. Put n = 1 in 3n and find what is left over.
   3 × 1 + ? = 1, so ? = -2 = -2
4. Write the nth term.
   3n − 2

Check. Check n = 2: 3n − 2 gives 4, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 3
2. Write the first part of the nth term. The nth term starts with the difference times n.
   3n
3. Find the number to add. The first term is 6. Put n = 1 in 3n and find what is left over.
   3 × 1 + ? = 6, so ? = 3 = 3
4. Write the nth term.
   3n + 3

Check. Check n = 2: 3n + 3 gives 9, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 2
2. Write the first part of the nth term. The nth term starts with the difference times n.
   2n
3. Find the number to add. The first term is 4. Put n = 1 in 2n and find what is left over.
   2 × 1 + ? = 4, so ? = 2 = 2
4. Write the nth term.
   2n + 2

Check. Check n = 2: 2n + 2 gives 6, which is the second term. ✓

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 5
2. Write the first part of the nth term. The nth term starts with the difference times n.
   5n
3. Find the number to add. The first term is 7. Put n = 1 in 5n and find what is left over.
   5 × 1 + ? = 7, so ? = 2 = 2
4. Write the nth term.
   5n + 2

Check. Check n = 2: 5n + 2 gives 12, which is the second term. ✓

Method. Use the nth term rule. Put in the position n and work it out.

1. Write the nth term. This is the rule for any term.
   nth term = 4n − 1
2. Put in n. We want term number 10, so put n = 10 into the rule.
   4 × 10 − 1
3. Work it out. Multiply first, then add or subtract.
   40 − 1 = 39
4. Write the answer.
   39

Check. Put in n = 9 to check the term before: 35, and 39 is 4 more.

Method. Use the nth term rule. Put in the position n and work it out.

1. Write the nth term. This is the rule for any term.
   nth term = 3n + 2
2. Put in n. We want term number 20, so put n = 20 into the rule.
   3 × 20 + 2
3. Work it out. Multiply first, then add or subtract.
   60 + 2 = 62
4. Write the answer.
   62

Check. Put in n = 19 to check the term before: 59, and 62 is 3 more.

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 47.
   4n − 1 = 47
2. Solve for n. Subtract 1 and divide by 4.
   n = (47 − -1) ÷ 4 = 12
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 12
4. Write the answer.
   Yes

Check. 4n − 1 gives 47 only when n = 12.

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 50.
   4n − 1 = 50
2. Solve for n. Subtract 1 and divide by 4.
   n = (50 − -1) ÷ 4 = 51/4
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 51/4
4. Write the answer.
   No

Check. Check n = 51/4 is not a whole number.

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 102.
   5n + 2 = 102
2. Solve for n. Subtract 2 and divide by 5.
   n = (102 − 2) ÷ 5 = 20
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 20
4. Write the answer.
   Yes

Check. 5n + 2 gives 102 only when n = 20.

Method. Use the nth term rule. Put in the position n and work it out.

1. Write the nth term. This is the rule for any term.
   nth term = 3n + 2
2. Put in n. We want term number 10, so put n = 10 into the rule.
   3 × 10 + 2
3. Work it out. Multiply first, then add or subtract.
   30 + 2 = 32
4. Write the answer.
   32

Check. Put in n = 9 to check the term before: 29, and 32 is 3 more.

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 100.
   6n + 1 = 100
2. Solve for n. Subtract 1 and divide by 6.
   n = (100 − 1) ÷ 6 = 33/2
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 33/2
4. Write the answer.
   No

Check. Check n = 33/2 is not a whole number.

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

1. Look at the last two terms. Add the last two terms to find the next one.
   5 and 8
2. Add them. The next term is the sum of these two.
   5 + 8 = 13
3. Repeat for the next term. Add the two terms before the new one.
   5 + 8 = 13 ; 8 + 13 = 21
4. Write the answer.
   13, 21

Check. Check: each new term equals the sum of the two before it.

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

1. Look at the last two terms. Add the last two terms to find the next one.
   8 and 13
2. Add them. The next term is the sum of these two.
   8 + 13 = 21
3. Repeat for the next term. Add the two terms before the new one.
   8 + 13 = 21 ; 13 + 21 = 34
4. Write the answer.
   21, 34

Check. Check: each new term equals the sum of the two before it.

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

1. Look at the last two terms. Add the last two terms to find the next one.
   11 and 18
2. Add them. The next term is the sum of these two.
   11 + 18 = 29
3. Repeat for the next term. Add the two terms before the new one.
   11 + 18 = 29
4. Write the answer.
   29

Check. Check: each new term equals the sum of the two before it.

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

1. Look at the last two terms. Add the last two terms to find the next one.
   12 and 19
2. Add them. The next term is the sum of these two.
   12 + 19 = 31
3. Repeat for the next term. Add the two terms before the new one.
   12 + 19 = 31
4. Write the answer.
   31

Check. Check: each new term equals the sum of the two before it.

Method. Divide each term by the one before it. If the answer is always the same, multiply by that number each time.

1. Divide the second term by the first. This gives the multiplier.
   6 ÷ 2 = 3
2. Check the other terms. The same multiplier must work for each pair.
   6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3
3. Use the rule to find the next term. Multiply the last term by 3.
   54 × 3 = 162
4. Write the answer.
   162, 486

Check. The rule is multiply by 3. Each term is 3 times the term before it.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   3 × 2^{n − 1}
2. Put in n. We want term 6, so n − 1 = 5.
   3 × 2⁵
3. Work it out. Multiply by r once for each step.
   3 × 32 = 96
4. Write the answer.
   96

Check. Check by multiplying step by step: 3 × 2 × 2 × 2 × 2 × 2 = 96.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   2 × 3^{n − 1}
2. Put in n. We want term 5, so n − 1 = 4.
   2 × 3⁴
3. Work it out. Multiply by r once for each step.
   2 × 81 = 162
4. Write the answer.
   162

Check. Check by multiplying step by step: 2 × 3 × 3 × 3 × 3 = 162.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   5 × -2^{n − 1}
2. Put in n. We want term 4, so n − 1 = 3.
   5 × -2³
3. Work it out. Multiply by r once for each step.
   5 × -8 = -40
4. Write the answer.
   −40

Check. Check by multiplying step by step: 5 × -2 × -2 × -2 = -40.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   200 × 2^{n − 1}
2. Put in n. We want term 4, so n − 1 = 3.
   200 × 2³
3. Work it out. Multiply by r once for each step.
   200 × 8 = 1600
4. Write the answer.
   1600

Check. Check by multiplying step by step: 200 × 2 × 2 × 2 = 1600.

Method. Write the terms of the sequence, and see whether 25 appears.

1. Write the sequence. The ratio is 2, so the terms are 5, 10, 20, 40, 80, ...
   5, 10, 20, 40, 80
2. Compare with 25. 25 is not in this list. The terms jump from 20 to 40.
   25 ≠ 20 and 25 ≠ 40
3. Write the answer. So 25 is not a term.
   No

Check. The ratio is 2, so no term can be 25.

Method. Check whether there is a constant ratio.

1. Find the ratios. 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 5 ÷ 3 = 1.67
   2, 1.5, 1.67
2. Compare. The ratios are not the same.
   2 ≠ 1.5
3. Write the answer. This is a Fibonacci-type sequence, not a geometric one. Ali is not right.
   No

Check. The next Fibonacci-type term is 8 + 5 = 13, not a multiple of 8.

Method. Find the ratio, then write the next term.

1. Find the ratio. 4 ÷ 2 = 2 and 8 ÷ 4 = 2.
   ratio = 2
2. Find the next term. Multiply the last height by 2.
   8 × 2 = 16
3. Write the answer. The sequence is geometric with ratio 2.
   16 cm

Check. 2, 4, 8, 16 each doubles the one before.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = -1. Put -1 in place of x in y = 2x + 1.
   y = 2 × (−1) + 1 = -1
2. Substitute x = 0. Put 0 in place of x in y = 2x + 1.
   y = 2 × (0) + 1 = 1
3. Substitute x = 1. Put 1 in place of x in y = 2x + 1.
   y = 2 × (1) + 1 = 3
4. Substitute x = 2. Put 2 in place of x in y = 2x + 1.
   y = 2 × (2) + 1 = 5
5. Substitute x = 3. Put 3 in place of x in y = 2x + 1.
   y = 2 × (3) + 1 = 7
6. Write the table. Each y value goes with its x value.
   (-1, -1), (0, 1), (1, 3), (2, 5), (3, 7)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = 0. Put 0 in place of x in y = 4x − 1.
   y = 4 × (0) − 1 = -1
2. Substitute x = 1. Put 1 in place of x in y = 4x − 1.
   y = 4 × (1) − 1 = 3
3. Substitute x = 2. Put 2 in place of x in y = 4x − 1.
   y = 4 × (2) − 1 = 7
4. Substitute x = 3. Put 3 in place of x in y = 4x − 1.
   y = 4 × (3) − 1 = 11
5. Write the table. Each y value goes with its x value.
   (0, -1), (1, 3), (2, 7), (3, 11)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = 0. Put 0 in place of x in y = −x + 5.
   y = -1 × (0) + 5 = 5
2. Substitute x = 1. Put 1 in place of x in y = −x + 5.
   y = -1 × (1) + 5 = 4
3. Substitute x = 2. Put 2 in place of x in y = −x + 5.
   y = -1 × (2) + 5 = 3
4. Substitute x = 3. Put 3 in place of x in y = −x + 5.
   y = -1 × (3) + 5 = 2
5. Write the table. Each y value goes with its x value.
   (0, 5), (1, 4), (2, 3), (3, 2)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = -1. Put -1 in place of x in y = 3x − 2.
   y = 3 × (−1) − 2 = -5
2. Substitute x = 0. Put 0 in place of x in y = 3x − 2.
   y = 3 × (0) − 2 = -2
3. Substitute x = 1. Put 1 in place of x in y = 3x − 2.
   y = 3 × (1) − 2 = 1
4. Substitute x = 2. Put 2 in place of x in y = 3x − 2.
   y = 3 × (2) − 2 = 4
5. Write the table. Each y value goes with its x value.
   (-1, -5), (0, -2), (1, 1), (2, 4)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. The points from the table are (−1, −1), (0, 1), (1, 3), (2, 5) and (3, 7). Plot them and join with a line.

1. Use the table. The points are the pairs (x, y) from the table.
   (−1, −1), (0, 1), (1, 3), (2, 5), (3, 7)
2. Plot the points. Move across by x, then up by y.
   five points
3. Draw the line. Join the points with a ruler. They lie on a straight line.
   straight line

Check. The line crosses the y-axis at (0, 1).

Method. Put each x value into the rule and check the y value.

1. Check (0, 1). 2 × 0 + 1 = 1, which matches y = 1.
   1 = 1, so yes
2. Check (2, 6). 2 × 2 + 1 = 5, but y = 6, so no.
   5 ≠ 6
3. Check (3, 7). 2 × 3 + 1 = 7, which matches y = 7.
   7 = 7, so yes
4. Write the answer. The points on the line are (0, 1) and (3, 7).
   (0, 1) and (3, 7)

Check. Only the points that satisfy the rule are on the line.

Method. Put x = 3 into the rule and see if y = 8.

1. Put x = 3 into the rule. 2 × 3 + 2 = 6 + 2.
   6 + 2 = 8
2. Compare with y. The y value in the point is 8.
   8 = 8
3. Write the answer. So the point is on the line.
   Yes

Check. (3, 8) is on the line, so it satisfies y = 2x + 2.

Method. Put x = 2 into the rule.

1. Put x = 2 into the rule. 3 × 2 − 2 = 6 − 2.
   6 − 2 = 4
2. Compare. y should be 4, not 3.
   4 ≠ 3
3. Write the answer. Ali is not right.
   No

Check. The point (2, 4) lies on the line, but (2, 3) does not.

Method. Put x = 2 into the rule.

1. Put x = 2 into y = 3x + 1. 3 × 2 + 1 = 7
   3 × 2 + 1 = 7
2. Write the point. The point is (2, 7).
   (2, 7)

Check. The point is on the line, because it follows the rule.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   9 − 3 = 6
2. Find the change in x. Subtract the x values, in the same order.
   4 − 1 = 3
3. Divide. Gradient = change in y ÷ change in x.
   6 ÷ 3 = 2
4. Write the answer.
   2

Check. Check with the other way round: -6 ÷ -3 = 2. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   1 − 5 = -4
2. Find the change in x. Subtract the x values, in the same order.
   2 − 0 = 2
3. Divide. Gradient = change in y ÷ change in x.
   -4 ÷ 2 = −2
4. Write the answer.
   −2

Check. Check with the other way round: 4 ÷ -2 = −2. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   5 − -3 = 8
2. Find the change in x. Subtract the x values, in the same order.
   3 − -1 = 4
3. Divide. Gradient = change in y ÷ change in x.
   8 ÷ 4 = 2
4. Write the answer.
   2

Check. Check with the other way round: -8 ÷ -4 = 2. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   9 − 1 = 8
2. Find the change in x. Subtract the x values, in the same order.
   6 − 2 = 4
3. Divide. Gradient = change in y ÷ change in x.
   8 ÷ 4 = 2
4. Write the answer.
   2

Check. Check with the other way round: -8 ÷ -4 = 2. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   2 − 7 = -5
2. Find the change in x. Subtract the x values, in the same order.
   5 − 0 = 5
3. Divide. Gradient = change in y ÷ change in x.
   -5 ÷ 5 = −1
4. Write the answer.
   −1

Check. Check with the other way round: 5 ÷ -5 = −1. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   10 − 4 = 6
2. Find the change in x. Subtract the x values, in the same order.
   3 − 1 = 2
3. Divide. Gradient = change in y ÷ change in x.
   6 ÷ 2 = 3
4. Write the answer.
   3

Check. Check with the other way round: -6 ÷ -2 = 3. The gradient is the same.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   -3 − 1 = -4
2. Find the change in x. Subtract the x values, in the same order.
   2 − -2 = 4
3. Divide. Gradient = change in y ÷ change in x.
   -4 ÷ 4 = −1
4. Write the answer.
   −1

Check. Check with the other way round: 4 ÷ -4 = −1. The gradient is the same.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = 3x − 2
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = 3, c = -2
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   -2

Check. Put x = 0: y = 3 × 0 − 2 = -2.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = 4x + 7
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = 4, c = 7
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   7

Check. Put x = 0: y = 4 × 0 + 7 = 7.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = −x + 3
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = -1, c = 3
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   3

Check. Put x = 0: y = -1 × 0 + 3 = 3.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = 2x
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = 2, c = 0
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   0

Check. Put x = 0: y = 2 × 0 + 0 = 0.

Method. Parallel lines have the same gradient. Find the gradient of each line.

1. Find the gradient of y = 2x + 5. The number in front of x is 2.
   m = 2
2. Compare the gradients. Line (a) has gradient 2, line (b) has gradient 1, and line (c) has gradient 5.
   a: 2, b: 1, c: 5
3. Write the answer. Only line (a) has the same gradient.
   (a) y = 2x − 3

Check. The lines y = 2x + 5 and y = 2x − 3 never meet.

Method. Compare the gradients.

1. Find the gradients. The gradient of y = 3x + 4 is 3. The gradient of y = −3x + 4 is −3.
   3 and −3
2. Compare. The gradients are different, so the lines are not parallel.
   3 ≠ −3
3. Write the answer. Ali is not right.
   No

Check. The lines cross where 3x + 4 = −3x + 4, which is at x = 0.

Method. Find the gradient first. The point (0, 4) gives the y-intercept.

1. Find the gradient. Change in y = 10 − 4 = 6. Change in x = 2 − 0 = 2.
   6 ÷ 2 = 3
2. Find the y-intercept. The point (0, 4) has x = 0, so c = 4.
   c = 4
3. Write the equation. Put m = 3 and c = 4 into y = mx + c.
   y = 3x + 4

Check. Check (2, 10): 3 × 2 + 4 = 10.

Method. Put x = 4 into the rule, and remember the starting charge.

1. Put x = 4 into the rule. The rule is y = 1.5x + 2.
   1.5 × 4 + 2
2. Multiply. The distance charge for 4 miles is 6.
   1.5 × 4 = 6
3. Add the starting charge. Add the fixed £2.
   6 + 2 = 8
4. Write the answer.
   £8.00

Check. The gradient 1.5 is the cost per mile. The intercept 2 is the starting charge.

Method. Read the gradient and the intercept from the rule.

1. Find the gradient. The number in front of x is 1.5.
   m = 1.5
2. Explain the gradient. The line goes up £1.50 for each mile, so 1.5 is the cost per mile.
   £1.50 per mile
3. Find the y-intercept. The number on its own is 2. This is the fare when x = 0, the starting charge.
   c = 2
4. Write the answer.
   Gradient 1.5 is the cost per mile. The intercept 2 is the starting charge.

Check. At 0 miles the fare is £2.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   5 − -3 = 8
2. Find the change in x. Subtract the x values, in the same order.
   3 − -1 = 4
3. Divide. Gradient = change in y ÷ change in x.
   8 ÷ 4 = 2
4. Write the answer.
   2

Check. Check with the other way round: -8 ÷ -4 = 2. The gradient is the same.

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   11 − 7 = 4
2. Check the other differences. Every difference must be the same for the rule to work.
   11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4
3. Use the rule to find the next term. Add 4 to the last term.
   19 + 4 = 23
4. Write the answer.
   23, 27

Check. The rule is add 4. Each term is 4 more than the one before.

Method. Divide each term by the one before it. If the answer is always the same, multiply by that number each time.

1. Divide the second term by the first. This gives the multiplier.
   48 ÷ 96 = 1/2
2. Check the other terms. The same multiplier must work for each pair.
   48 ÷ 96 = 1/2, 24 ÷ 48 = 1/2
3. Use the rule to find the next term. Multiply the last term by 1/2.
   24 × 1/2 = 12
4. Write the answer.
   12, 6

Check. The rule is multiply by 1/2. Each term is 1/2 times the term before it.

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = 5
2. Write the first part of the nth term. The nth term starts with the difference times n.
   5n
3. Find the number to add. The first term is 4. Put n = 1 in 5n and find what is left over.
   5 × 1 + ? = 4, so ? = -1 = -1
4. Write the nth term.
   5n − 1

Check. Check n = 2: 5n − 1 gives 9, which is the second term. ✓

Method. Use the nth term rule. Put in the position n and work it out.

1. Write the nth term. This is the rule for any term.
   nth term = 2n + 5
2. Put in n. We want term number 12, so put n = 12 into the rule.
   2 × 12 + 5
3. Work it out. Multiply first, then add or subtract.
   24 + 5 = 29
4. Write the answer.
   29

Check. Put in n = 11 to check the term before: 27, and 29 is 2 more.

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 61.
   4n + 1 = 61
2. Solve for n. Subtract 1 and divide by 4.
   n = (61 − 1) ÷ 4 = 15
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 15
4. Write the answer.
   Yes

Check. 4n + 1 gives 61 only when n = 15.

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

1. Look at the last two terms. Add the last two terms to find the next one.
   7 and 11
2. Add them. The next term is the sum of these two.
   7 + 11 = 18
3. Repeat for the next term. Add the two terms before the new one.
   7 + 11 = 18
4. Write the answer.
   18

Check. Check: each new term equals the sum of the two before it.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   2 × 3^{n − 1}
2. Put in n. We want term 5, so n − 1 = 4.
   2 × 3⁴
3. Work it out. Multiply by r once for each step.
   2 × 81 = 162
4. Write the answer.
   162

Check. Check by multiplying step by step: 2 × 3 × 3 × 3 × 3 = 162.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = 0. Put 0 in place of x in y = 2x − 3.
   y = 2 × (0) − 3 = -3
2. Substitute x = 1. Put 1 in place of x in y = 2x − 3.
   y = 2 × (1) − 3 = -1
3. Substitute x = 2. Put 2 in place of x in y = 2x − 3.
   y = 2 × (2) − 3 = 1
4. Substitute x = 3. Put 3 in place of x in y = 2x − 3.
   y = 2 × (3) − 3 = 3
5. Write the table. Each y value goes with its x value.
   (0, -3), (1, -1), (2, 1), (3, 3)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   10 − 2 = 8
2. Find the change in x. Subtract the x values, in the same order.
   3 − 1 = 2
3. Divide. Gradient = change in y ÷ change in x.
   8 ÷ 2 = 4
4. Write the answer.
   4

Check. Check with the other way round: -8 ÷ -2 = 4. The gradient is the same.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = 5x + 1
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = 5, c = 1
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   1

Check. Put x = 0: y = 5 × 0 + 1 = 1.

Method. Find the difference between each pair of terms. If it is always the same, that is the rule.

1. Find the difference between the first two terms. Subtract the first term from the second.
   8 − 3 = 5
2. Check the other differences. Every difference must be the same for the rule to work.
   8 − 3 = 5, 13 − 8 = 5, 18 − 13 = 5
3. Use the rule to find the next term. Add 5 to the last term.
   18 + 5 = 23
4. Write the answer.
   23, 28

Check. The rule is add 5. Each term is 5 more than the one before.

Method. Put n = 1, 2 and 3 into the rule.

1. Put n = 1. 5 × 1 + 3 = 8.
   8
2. Put n = 2. 5 × 2 + 3 = 13.
   13
3. Put n = 3. 5 × 3 + 3 = 18.
   18
4. Write the answer.
   8, 13, 18

Check. Each term is 5 more than the one before.

Method. The difference is constant, so the nth term is d n + (first term − d). Then check it works for the first terms.

1. Find the common difference. Subtract each term from the next.
   d = -3
2. Write the first part of the nth term. The nth term starts with the difference times n.
   -3n
3. Find the number to add. The first term is 10. Put n = 1 in -3n and find what is left over.
   -3 × 1 + ? = 10, so ? = 13 = 13
4. Write the nth term.
   −3n + 13

Check. Check n = 2: −3n + 13 gives 7, which is the second term. ✓

Method. Put the value equal to the nth term, then solve for n. If n is a whole number, the value is in the sequence.

1. Write the equation. Set the nth term equal to 81.
   3n + 2 = 81
2. Solve for n. Subtract 2 and divide by 3.
   n = (81 − 2) ÷ 3 = 79/3
3. Decide. n must be a whole number, positive, for it to be a term.
   n = 79/3
4. Write the answer.
   No

Check. Check n = 79/3 is not a whole number.

Method. Each term is the one before multiplied by the same number. Use the rule: term n = first term × r^{n − 1}.

1. Write the rule. The first term is multiplied by r for each step after it.
   5 × 2^{n − 1}
2. Put in n. We want term 7, so n − 1 = 6.
   5 × 2⁶
3. Work it out. Multiply by r once for each step.
   5 × 64 = 320
4. Write the answer.
   320

Check. Check by multiplying step by step: 5 × 2 × 2 × 2 × 2 × 2 × 2 = 320.

Method. Put each x value into the rule, then work out y. Multiply first, then add.

1. Substitute x = -1. Put -1 in place of x in y = 3x − 1.
   y = 3 × (−1) − 1 = -4
2. Substitute x = 0. Put 0 in place of x in y = 3x − 1.
   y = 3 × (0) − 1 = -1
3. Substitute x = 1. Put 1 in place of x in y = 3x − 1.
   y = 3 × (1) − 1 = 2
4. Substitute x = 2. Put 2 in place of x in y = 3x − 1.
   y = 3 × (2) − 1 = 5
5. Write the table. Each y value goes with its x value.
   (-1, -4), (0, -1), (1, 2), (2, 5)

Check. Each point is on the straight line, because the y values change by the same amount each time.

Method. The equation is in the form y = mx + c. The number c is the y-intercept.

1. Write the equation in the form y = mx + c. Compare the equation with y = mx + c.
   y = −2x + 7
2. Find m and c. m is the number in front of x, and c is the number on its own.
   m = -2, c = 7
3. Write the answer. The line crosses the y-axis where x = 0, so y = c.
   7

Check. Put x = 0: y = -2 × 0 + 7 = 7.

Method. Gradient = change in y ÷ change in x. Use the two points.

1. Find the change in y. Subtract the y values.
   11 − 5 = 6
2. Find the change in x. Subtract the x values, in the same order.
   3 − 1 = 2
3. Divide. Gradient = change in y ÷ change in x.
   6 ÷ 2 = 3
4. Write the answer.
   3

Check. Check with the other way round: -6 ÷ -2 = 3. The gradient is the same.

Method. Parallel lines have the same gradient.

1. Find the gradient of y = 4x − 2. The gradient is 4.
   m = 4
2. Compare with each line. Line (a) has gradient 4, line (b) has 2 and line (c) has −4.
   a: 4, b: 2, c: −4
3. Write the answer. Only line (a) has gradient 4.
   (a) y = 4x + 3

Check. y = 4x − 2 and y = 4x + 3 are parallel.

Method. Put x = 6 into the rule.

1. Put x = 6 into the rule. The rule is y = 2x + 5.
   2 × 6 + 5
2. Work it out. Multiply, then add.
   12 + 5 = 17
3. Write the answer.
   £17.00

Check. The £5 is the joining fee, and £12 is 6 visits at £2.