Year 7 · Unit 3 Patterns, expressions and equations · Lesson 3 of 6

How do you find the 100th term without listing 99?

A term-to-term rule needs the term before. A position rule jumps straight to any term. Slide the first term and the step and watch both rules come from the same pattern.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Unit 3, lesson 1, growing patterns. What is the gap in 12, 19, 26, 33?
    Answer7. Each term is 7 more than the one before.
  2. From Unit 3, lesson 2, letters and expressions. Work out 4n − 3 when n = 5.
    Answer17. 4 × 5 = 20, then 20 − 3.
  3. From Unit 1, factors and multiples. What is the 9th multiple of 7?
    Answer63. 9 × 7 = 63.
Words used in this lesson
term-to-term rule
says how to get from one term to the next. For example: add 4.
position rule
gives a term from its position n. For example: 4n + 1.
position
the place of a term in the sequence, counting from 1. For example: 10 is in position 2 of 5, 10, 15.
step
the amount added each time. For example: 4 in 5, 9, 13.
linear sequence
a sequence with the same step every time. For example: 3, 7, 11, 15.
test
check a rule on a case you already know. For example: 4 × 1 + 1 = 5.

Two ways to describe the same pattern

Row 1 of a stadium stand has 5 seats and every row has 4 more, so the rows hold 5, 9, 13, 17, … seats. How many seats are in row 100? You can describe 5, 9, 13, 17, … in two ways. The term-to-term rule says how to get from one term to the next: add 4. It is quick for the next term, but to find the 100th term you would have to know the 99th. The position rule says how to get a term straight from its position n, so any term comes with no list. How many seats are in row 100?

Predict first: the first term is 5 and each term is 4 more. Before you move the sliders, what is the number that goes with n in the position rule? Then check it with the step.

5
4
10
add 4term-to-term rule
4n + 1position rule
41the chosen term, term k

Each term is 4 more than the one before. Position rule: 4n + 1. Term 10: 4 × 10 + 1 = 41.

Show me the steps
  1. First term 5 and step 4: 5, 9, 13, 17, 21, 25. The term-to-term rule is add 4. The position rule is 4n + 1, and term 10 is 4 × 10 + 1 = 41.
  2. Test the position rule on the first terms: 4 × 1 + 1 = 5 and 4 × 2 + 1 = 9. Both match the bars.
  3. First term 2 and step 3: 2, 5, 8, 11, 14, 17. The position rule is 3n − 1, because 3 × 1 − 1 = 2. Term 10 is 3 × 10 − 1 = 29.
  4. First term 3 and step 7: 3, 10, 17, 24, 31, 38. The position rule is 7n − 4, and term 30 is 7 × 30 − 4 = 206.
  5. First term 5 and step 5: the terms are the 5 times table, 5, 10, 15, and the position rule is just 5n.

The number in front of n is the step. The term-to-term rule is add 4, so the position rule starts with 4n. Then fix the first term: 4 × 1 = 4, but the first term is 5, so add 1. The position rule is 4n + 1. Always test it on the first terms. The number at the end is the term that would come before the first, term 0: 5 − 4 = 1, so 4n + 1.

A sequence is linear if the step is the same every time. The position rule of a linear sequence has the form (step)n plus or minus a number.

Back to the stand: the rule is 4n + 1, so row 100 holds 4 × 100 + 1 = 401 seats, with no list.

Is this number in the sequence?

To test whether a number belongs to a sequence, undo the steps. Take away the first term, divide by the step, and add 1. If the division is exact, the number is a term and the result is its position.

3
7
164
yesis N a term?
24its term number

164 is a term: 164 − 3 = 161, 161 ÷ 7 = 23 steps, so it is term 24.

Show me the steps
  1. 3, 10, 17, 24, … and N = 164: 164 − 3 = 161, 161 ÷ 7 = 23 steps, and 23 + 1 = 24. It is the 24th term.
  2. N = 165: 165 − 3 = 162, and 162 ÷ 7 is about 23.14, not a whole number. It is not a term; it lies between the terms 164 and 171.
  3. N = 2 is smaller than the first term, 3, so it is not in the sequence at all.
  4. First term 4 and step 5, N = 99: 99 − 4 = 95, 95 ÷ 5 = 19 steps, so it is the 20th term.
  5. First term 4 and step 5, N = 100: 100 − 4 = 96, and 96 ÷ 5 = 19.2. Not a term; it lies between 99 and 104.

Undoing the position rule gives the same answer. For 3, 10, 17, 24, … the rule is 7n − 4. To test 164: 164 = 7n − 4, add 4 to get 168, divide by 7 to get n = 24.

A common mistake: “The rule is 3n, because it goes up by 3.” Try n = 1. The sequence 7, 10, 13 has first term 7, but 3n gives 3. The step is only the first part of the rule. Then fix the first term: 3n + 4.

Worked example: the 40th term

Find the 40th term of 9, 14, 19, 24, …

  1. The step is 5, so the position rule starts with 5n.
  2. For n = 1, 5 × 1 = 5, but the first term is 9, which is 4 more: the rule is 5n + 4.
  3. Test it: 5 × 2 + 4 = 14, which is the second term.
  4. For the 40th term: 5 × 40 + 4 = 204.

Your go. The first step is given. Find the 25th term of 2, 8, 14, 20, …

  1. The step is 6, so the position rule starts with 6n.
  2. Now find what to add or subtract to get the first term, test your rule, then find the 25th term yourself.
Check the answer

6 × 1 = 6, and the first term is 2, so subtract 4: the rule is 6n − 4. Test: 6 × 2 − 4 = 8. The 25th term is 6 × 25 − 4 = 150 − 4 = 146.

Check before you go on

1. Which is the position rule for the sequence 6, 10, 14, 18, …?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The position rule of a sequence is 5n + 3. What is the 4th term?

3. A sequence starts 2, 9, 16, 23, … and the rule is add 7. What is the 6th term?

Level 2 · Building

4. The seats in the rows of a stand go 3, 7, 11, 15, … for rows 1, 2, 3, 4, and the position rule is 4n − 1. How many seats are in row 50?

5. Which term of 7, 12, 17, 22, … is equal to 117? The position rule is 5n + 2.

6. Which of these numbers is a term of 5, 11, 17, 23, …? The position rule is 6n − 1.

Level 3 · Stretch

7. The sequence 6, 13, 20, 27, … has position rule 7n − 1. What is the first term greater than 200?

8. A linear sequence has 5th term 23 and 6th term 28. What is its 1st term?

Something to think about

Write the first four terms of the position rule 4n + 3, then the first four terms of 4n − 1. What do you notice, and why does it happen?

One way to think about it

4n + 3 gives 7, 11, 15, 19 and 4n − 1 gives 3, 7, 11, 15. They are the same list; the second just starts one term earlier. The term 4n − 1 is the term before 4n + 3, because 4n − 1 + 4 = 4n + 3.

Exit ticket

9. Ngozi says: “The rule for 7, 10, 13, 16, … is 3n, because it goes up by 3.” Which reply is right?