Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 1 of 8

How do you reach row 25 without writing out 24 rows?

A sequence that adds the same amount every step has a rule that jumps straight to any position. Where the rule comes from, and how to use it in both directions, is the work of this lesson.

Number and AlgebraGDC optional

Warm-up

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

  1. From Year 7: the sequence 5, 9, 13, 17 goes up by the same amount each time. What is the next term?
    Answer21. The terms go up by 4.
  2. From Year 9: a line has gradient 3 and passes through (0, 2). What is its equation?
    Answery = 3x + 2.
  3. From Year 8: evaluate 4n − 1 when n = 10.
    Answer39. 4 × 10 = 40, then take away 1.
Words used in this lesson
sequence
a list of numbers in a set order. For example: 2, 11, 20, 29.
term
one number in a sequence. For example: the 3rd term of 8, 15, 22.
common difference
the amount added (or taken away) from one term to the next. For example: 3, 14, 25, 36 has difference 11.
arithmetic sequence
a sequence with a common difference. For example: 50, 43, 36, 29.
nth term
a rule that gives the term in position n. For example: 9n − 4.
position
the place of a term in the list, counted from 1. For example: n = 12.

How many seats in row 25?

The new cinema in Nairobi has 6 seats in the front row. Every row behind it has 4 more seats than the row in front. The manager needs the number of seats in row 25 to order the covers.

Predict first: would you write out all 25 rows, or is there a shortcut? Write down your estimate for row 25 and how you got it. Then use the figure: set the first term and the difference, and watch where the six terms sit.

2
2
2common difference
2nrule for the nth term

First term 2 and difference 2: the terms are 2, 4, 6, 8, 10, 12 and the rule is 2n.

Show me the steps
  1. a = 2, d = 2: the terms are 2, 4, 6, 8, 10, 12. Each one is twice its position, so the rule is 2n.
  2. a = 2, d = 5: the terms are 2, 7, 12, 17, 22, 27. The points are steeper, and the rule is 5n − 3: the constant is a − d = 2 − 5.
  3. a = 9, d = −3: the terms are 9, 6, 3, 0, −3, −6. The points fall, and the rule is 12 − 3n.
  4. Any a and d: the nth term is a + (n − 1)d = dn + (a − d). The gradient of the line is d, and the constant is the value the line would take at n = 0.

The nth term of an arithmetic sequence is dn + c. A sequence with common difference d has terms that go up (or down) by the same amount each step. To reach position n from the first term a you take n − 1 steps: a + (n − 1)d. This simplifies to dn + (a − d), a straight-line rule in n.

Check the rule at n = 1: it must return the first term. That one test catches most slips.

Two common mistakes. Using the first term as the constant: the sequence 15, 23, 31 has rule 8n + 7, not 8n + 15, because at n = 1 the rule gives 8 + 7 = 15. And counting n steps instead of n − 1: from term 1 to term 8 there are 7 jumps, not 8.

Worked examples

1. Find the nth term of 5, 8, 11, 14, …

  1. The differences are 8 − 5 = 3, 11 − 8 = 3 and 14 − 11 = 3, so d = 3 and the rule starts 3n.
  2. At n = 1, 3n gives 3 and the first term is 5, so the constant is 5 − 3 = 2. The rule is 3n + 2. Check n = 4: 12 + 2 = 14. ✓

2. In the sequence with nth term 3n + 2, which term is 98?

  1. Set 3n + 2 = 98, so 3n = 96.
  2. n = 96 ÷ 3 = 32. It is the 32nd term. A value is a term only if n comes out as a positive whole number.

3. Find the nth term of 40, 34, 28, 22, …

  1. The terms go down by 6, so d = −6 and the rule starts −6n.
  2. At n = 1, −6n gives −6 and the first term is 40, so the constant is 40 − (−6) = 46. The rule is 46 − 6n. Check n = 3: 46 − 18 = 28. ✓

Your go. The first step is given. Find the nth term of 12, 17, 22, 27, … The common difference is 5, so the rule starts 5n.

  1. At n = 1, 5n gives 5.
  2. Now find the constant by comparing 5 with the first term, then write the rule and check it at n = 3.
Check the answer

The first term is 12, so the constant is 12 − 5 = 7 and the rule is 5n + 7. Check n = 3: 15 + 7 = 22. ✓

With a GDC (optional)

A GDC can list a sequence. Open the table or list menu, enter the rule with the variable X, and set the table to start at 1 and step by 1. The table then shows the terms in order, which is a quick way to test a rule you have found.

A table only shows the terms it lists, so it cannot replace the rule: to find the 200th term you still need the nth term. Everything here can be done by hand.

Check before you go on

1. A sequence starts 9, 16, 23, 30. Which is its nth term?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. An arithmetic sequence has first term 11 and common difference 6. What is the 9th term?

3. The nth term of a sequence is 8n − 5. Which term of the sequence is equal to 67? Give the position n.

Level 2 · Building

4. In an arithmetic sequence the 2nd term is 9 and the 10th term is 49. What is the common difference?

5. In an arithmetic sequence the 4th term is 35 and the 9th term is 10. What is the first term?

Level 3 · Stretch

6. One sequence has nth term 6n + 1 and another has nth term 9n − 20. For which position n are the two terms equal?

7. The sequence 100, 94, 88, 82, … keeps falling by 6. How many of its terms are greater than 0?

Something to think about

A sequence has nth term 4n + 2. Another has nth term 4n + 3. Can the two sequences ever share a term? Test the first few terms of each, then explain using remainders on division by 4.

One way to think about it

No. Every term of the first sequence is 2 more than a multiple of 4, and every term of the second is 3 more than a multiple of 4, so no number can be in both. In general, two sequences with the same difference d share a term only when their constants differ by a multiple of d, and then one is the other with its first few terms removed.

Exit ticket

8. Which statement about the sequence with nth term 7 − 4n is correct?