Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 2 of 8

What is left when a pattern's differences are not constant?

When the gaps between terms are not equal, a pattern can still be hiding in them. This lesson looks one level deeper, at how the gaps themselves change, and what that says about the rule.

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 Year 11, lesson 1: find the common difference of 17, 24, 31, 38.
    Answer7.
  2. From Year 10: expand (n + 1)².
    Answern² + 2n + 1.
  3. From Year 8: evaluate 3n² when n = 4.
    Answer48. Square first: 16, then times 3.
Words used in this lesson
first difference
a term minus the term before it. For example: from 8, 15, 24: 7 and 9.
second difference
the difference between neighbouring first differences. For example: from 7 and 9: 2.
constant
staying the same every time. For example: the 5 in 5n + 1.
quadratic sequence
a sequence whose nth term has an n squared term. For example: n² + 4.
coefficient
the number multiplying a letter. For example: the 9 in 9n².

Why do the tiles keep going up by different amounts?

A tiler in Kumasi lays patios in a growing pattern. The first three patios use 3, 7 and 13 tiles, and the fourth and fifth use 21 and 31. It is not an arithmetic sequence, because the gaps between the numbers of tiles are not equal.

Predict first: work out the gaps between the five numbers, and then the gaps between those gaps. What do you notice, and what would the sixth patio need? Then use the figure to build other sequences with the same behaviour.

1
0
0
2second difference
1half of it

a = 1, b = 0, c = 0: terms 1, 4, 9, 16, 25, 36. The first differences 3, 5, 7, 9, 11 rise by 2, so the second difference is 2.

Show me the steps
  1. a = 1, b = 0, c = 0: the terms are 1, 4, 9, 16, 25, 36. The first differences are 3, 5, 7, 9, 11 and the second difference is 2.
  2. a = 4, b = −2, c = 3: the terms are 5, 15, 33, 59, 93, 135. The first differences are 10, 18, 26, 34, 42 and the second difference is 8, which is 2 × 4.
  3. a = 2, b = 3, c = 0: the terms are 5, 14, 27, 44, 65, 90. The second difference is 4. Changing b or c shifts the terms but not the second difference.
  4. Any a, b and c: the second difference is always 2a, so the number in front of n² is half the second difference.

Quadratic sequences have a constant second difference. Take the differences between the terms (the first differences), then the differences between those (the second differences). If the second differences are all equal and not zero, the nth term is an² + bn + c and the second difference equals 2a.

Why: term n + 1 minus term n is a(n + 1)² + b(n + 1) + c − (an² + bn + c) = 2an + a + b. That is a linear rule in n with difference 2a, so the first differences rise by 2a every step, whatever b and c are. So the number in front of n² is half the second difference.

Two common mistakes. Stopping at the first differences: 5, 9, 13 are not equal, but that does not make the sequence unpredictable. And using the second difference as the number in front of n²: a second difference of 12 means 6n², not 12n².

Worked examples

1. Is 1, 6, 15, 28, … a quadratic sequence?

  1. First differences: 6 − 1 = 5, 15 − 6 = 9, 28 − 15 = 13. They are not constant.
  2. Second differences: 9 − 5 = 4 and 13 − 9 = 4. They are constant, so it is quadratic.

2. A quadratic sequence has second difference 8. What number is in front of n²?

  1. The second difference is 2a, so 2a = 8.
  2. a = 8 ÷ 2 = 4. The nth term starts 4n².

3. Is 2, 5, 11, 23, … quadratic?

  1. First differences: 3, 6, 12. Second differences: 6 − 3 = 3 and 12 − 6 = 6.
  2. The second differences are not constant (3 then 6), so the sequence is not quadratic.

Your go. The first step is given. For 3, 8, 17, 30, 47, … the first differences are 5, 9, 13, 17. Find the second difference and the number in front of n².

  1. The second differences are 9 − 5 = 4, 13 − 9 = 4 and 17 − 13 = 4.
  2. Now halve the second difference to find the number in front of n².
Check the answer

The second difference is 4, so 2a = 4 and a = 2. The nth term starts 2n².

Check before you go on

1. The sequence 1, 5, 19, 43, 77 has first differences 4, 14, 24, 34. Which statement is correct?

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. A quadratic sequence begins 2, 9, 22, 41. The first differences are 7, 13, 19. What is the second difference?

3. A quadratic sequence has constant second difference 14. What number is in front of n² in its nth term?

Level 2 · Building

4. A sequence has nth term 5n² + 3n − 4. What is its second difference?

5. A sequence has first term 6 and first differences 5, 8, 11, 14, … (rising by 3 each time). What is the 6th term?

Level 3 · Stretch

6. The first differences of a quadratic sequence start at 7 and the second difference is 4. What is the difference between term 8 and term 9?

7. A quadratic sequence has second difference 6. Its first term is 3 and its second term is 10. What is the 4th term?

Something to think about

The first differences of a sequence are 2, 4, 8, 16, 32. The gaps of those gaps are 2, 4, 8, 16. What happens if you keep taking differences? Is the sequence quadratic?

One way to think about it

No. Every time you take differences you get the same kind of list again (doubling), so no row ever becomes constant. A quadratic sequence has constant second differences; this sequence grows by doubling gaps and belongs to a different family: a doubling sequence plus a constant, such as 1, 3, 7, 15, 31, 63, which is 2n − 1. Doubling is the geometric growth of lesson 4.

Exit ticket

8. Which of these sequences is quadratic?