Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From lesson 2: what is the second difference of 1, 4, 9, 16, 25?
Answer
2. The first differences 3, 5, 7, 9 rise by 2. - From lesson 1: find the nth term of 7, 11, 15, 19.
Answer
4n + 3. - From Year 8: simplify 5n² − 2n² + n.
Answer
3n² + n.
Words used in this lesson
- remainder
- what is left after subtracting the n squared part. For example: after taking away 4, 16, 36 from 5, 17, 37.
- halve
- divide by 2. For example: half of 18.
- nth term
- a rule that gives the term in position n. For example: 6n² + n.
- coefficient
- the number multiplying a letter. For example: the 8 in 8n.
- check
- testing a rule on terms you already know. For example: n = 2 in 3n² + 1.
Can you jump to the 20th patio?
A designer in Osaka lays rectangular patios of growing size in a garden path. The first four patios use 2, 9, 20 and 35 stones. The gaps between them are 7, 11 and 15, so the gaps of the gaps are constant. The client wants to know how many stones the 20th patio needs, and adding gaps one at a time would take a long time.
Predict first: the sequence curves, so its rule must have an n² term. What would you guess for the number in front of n², and what is left once that part is taken away? Then use the figure: pick a sequence and test guesses.
Sequence 1 with a guess of 1: the remainders are 5, 13, 23, 35, 49, 65 and their gaps 8, 10, 12, 14, 16 are not constant.
Show me the steps
- Sequence 1, guess 1: the terms 6, 17, 32, 51, 74, 101 minus 1, 4, 9, 16, 25, 36 leave 5, 13, 23, 35, 49, 65. The gaps 8, 10, 12, 14, 16 are not constant, so the guess is wrong.
- Sequence 1, guess 2: the remainders are 4, 9, 14, 19, 24, 29, which rise by 5 each time. The remainder is 5n − 1, so the sequence is 2n² + 5n − 1.
- Sequence 2, guess 3: the terms 1, 6, 17, 34, 57, 86 minus 3, 12, 27, 48, 75, 108 leave −2, −6, −10, −14, −18, −22, with a common difference of −4. The remainder is 2 − 4n.
- Any sequence: the guess that works is half the second difference, and the remainder is then an arithmetic sequence whose rule gives the bn + c.
Three steps for an² + bn + c. (1) Find the second difference and halve it: that is a. (2) Subtract an² from every term. (3) What is left is arithmetic: find its rule, bn + c, as in lesson 1.
Finish by checking the rule at n = 1 and n = 2 against the original terms.
Two common mistakes. Subtracting n² or an² from the terms but using the full second difference as a (a second difference of 8 means 4n²). And forgetting that the remainder is a rule in n: the remainders 9, 7, 5, 3 mean 11 − 2n, so the final rule is an² − 2n + 11, not an² + 11.
Worked examples
1. Find the nth term of 4, 15, 34, 61, …
- First differences 11, 19, 27; second difference 8; so a = 4 and the rule starts 4n².
- Subtract 4n² = 4, 16, 36, 64 from the terms: 0, −1, −2, −3. This falls by 1, and at n = 1 it is 0, so it is −n + 1.
- The nth term is 4n² − n + 1. Check n = 3: 36 − 3 + 1 = 34. ✓
2. Find the nth term of 8, 5, −2, −13, …
- First differences −3, −7, −11; second difference −4; so a = −2 and the rule starts −2n².
- Subtract −2n² = −2, −8, −18, −32, which means adding 2, 8, 18, 32: the remainders are 10, 13, 16, 19. They rise by 3 and start at 10, so they are 3n + 7.
- The nth term is −2n² + 3n + 7. Check n = 4: −32 + 12 + 7 = −13. ✓
3. Is 53 a term of the sequence n² + 4? Is 60?
- n² + 4 = 53 gives n² = 49, so n = 7 and 53 is the 7th term.
- n² + 4 = 60 gives n² = 56. That is not the square of a whole number, so 60 is not a term.
Your go. The first step is given. Find the nth term of 6, 14, 28, 48, … The first differences are 8, 14, 20 and the second difference is 6, so the rule starts 3n².
- Subtract 3n² = 3, 12, 27, 48 from the terms 6, 14, 28, 48.
- Find the rule for what is left, then write the nth term and check it at n = 3.
Check the answer
The remainders are 3, 2, 1, 0: they fall by 1 and start at 3, so they are 4 − n. The nth term is 3n² − n + 4. Check n = 3: 27 − 3 + 4 = 28. ✓
Check before you go on
1. The sequence 7, 12, 21, 34, 51 has second difference 4. After subtracting the n² part from each term, which list is left?
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. The terms 6, 15, 28, 45 have second difference 4. Subtracting 2n² from each term leaves an arithmetic sequence. What is its common difference?
3. A quadratic sequence has second difference 6 and, after subtracting 3n², a remainder of 4n − 1. What is the 4th term of the sequence?
Level 2 · Building
4. The sequence 7, 18, 35, 58 is quadratic. Write its nth term as an² + bn + c. What is b?
5. The sequence 5, 14, 29, 50, … is quadratic. Find its 10th term.
Level 3 · Stretch
6. A quadratic sequence has second difference 4, first term 5 and second term 16. Its nth term is 2n² + bn + c. What is c?
7. The nth term of a sequence is 2n² + 1. Which term of the sequence is 129? Give the position n.
Something to think about
Instead of taking away an², someone subtracts a number that is a little too big, such as 3n² when the true value is 2n². What do the remainders look like, and could you still recover the correct rule from them?
One way to think about it
The remainders are −n² + bn + c, so their second difference is −2 and the list is itself quadratic. That is the signal that the guess was wrong, and it tells you by how much: the guess was 1 too big, so taking 1 off it corrects it.
Exit ticket
8. Which is the nth term of 1, 7, 17, 31, 49?