Chapter 7 of 8

All 8 chapters
  1. 1Expanding two brackets, readunderstood
  2. 2Factorising quadratics, readunderstood
  3. 3Using factorising, readunderstood
  4. 4Completing the square, readunderstood
  5. 5The quadratic formula, readunderstood
  6. 6Graphs of quadratics, readunderstood
  7. 7Quadratic sequences, readunderstood
  8. 8A line and a curve, readunderstood

M4.9.7 Chapter 7

Quadratic sequences

The sequence 2, 6, 12, 20 goes up by more each time. Is there a rule for the nth term?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use second differences to spot a quadratic sequence
  • Find the nth term of a sequence like 2, 6, 12, 20

In an earlier book, a sequence like 3, 7, 11, 15 went up by 4 each time. Its first differences are all 4. That kind of sequence has an nth term like 4n − 1.

A row of the terms 3, 7, 11, 15 and 19 with their first differences under them: +4, +4, +4 and +4. Each difference sits halfway between the two terms it comes from.37111519+4+4+4+4
The first differences are all +4. This sequence goes up in equal steps.

Its second differences are all 0, because the steps never change.

Now take 2, 6, 12, 20. It goes up by 4, then 6, then 8. The differences are not equal. Look at the differences of the differences. They are the second differences.

Draw it first

First and second differences

  1. Step 1 shows the terms 2, 6, 12 and 20. Step 2 adds the first differences +4, +6 and +8 between them. Step 3 adds the second differences +2 and +2, with a dashed frame round them.261220

    1 Write the terms of the sequence in a row.

  2. Step 1 shows the terms 2, 6, 12 and 20. Step 2 adds the first differences +4, +6 and +8 between them. Step 3 adds the second differences +2 and +2, with a dashed frame round them.261220+4+6+8

    2 Subtract each term from the next one. These are the first differences: 4, 6 and 8.

  3. Step 1 shows the terms 2, 6, 12 and 20. Step 2 adds the first differences +4, +6 and +8 between them. Step 3 adds the second differences +2 and +2, with a dashed frame round them.261220+4+6+8+2+2

    3 Do it again with the first differences. The second differences are 2 and 2. They are the same.

When the second differences are all the same, and not 0, the sequence is a quadratic sequence. Its nth term has an n² in it. Why? Look at the square numbers, 1, 4, 9, 16, 25.

The terms 1, 4, 9, 16 and 25 with their first differences +3, +5, +7 and +9 under them, and the second differences +2, +2 and +2 under those, inside a dashed frame.1491625+3+5+7+9+2+2+2
The second differences of n² are all +2.

For n², the second difference is always 2. The first differences of n² are 3, 5, 7, 9, the odd numbers, and they go up by 2 each time. For 3n² it would be 3 × 2 = 6. For a n², the second difference is 2a. A part like bn + c goes up by the same b each step, so it adds nothing to the second differences. So the n² number is half the second difference.

Not every sequence has constant second differences. Take the cubes, 1, 8, 27, 64, 125. Their second differences change, so they are not quadratic.

The terms 1, 8, 27, 64 and 125 with their first differences +7, +19, +37 and +61, and the second differences +12, +18 and +24 under those. The second differences are not the same.182764125+7+19+37+61+12+18+24
The second differences, 12, 18 and 24, are not all the same. So the cubes are not a quadratic sequence.

Worked example

Find the nth term of 2, 6, 12, 20.

  1. The first differences are 4, 6, 8. The second differences are 2, 2. They are the same, so the sequence is quadratic.
  2. The second difference is 2, so the n² number is half of it: 2 ÷ 2 = 1. The nth term starts with n².
  3. The values of n² for n = 1, 2, 3, 4 are 1, 4, 9, 16.
  4. Take these from the sequence: 2 − 1 = 1, 6 − 4 = 2, 12 − 9 = 3 and 20 − 16 = 4.
  5. The results 1, 2, 3, 4 are the numbers n. So the nth term is n² + n.
  6. Check with n = 4: 16 + 4 = 20.

Answer: n² + n

Worked example

Find the nth term of 3, 8, 15, 24.

  1. The first differences are 5, 7, 9. The second differences are 2, 2. The sequence is quadratic.
  2. Start with n². Its number is half of the second difference: 2 ÷ 2 = 1.
  3. The values of n² are 1, 4, 9, 16.
  4. Take them from the sequence: 3 − 1 = 2, 8 − 4 = 4, 15 − 9 = 6 and 24 − 16 = 8.
  5. The results 2, 4, 6, 8 go up in 2s from 2. That is 2n.
  6. So the nth term is n² + 2n.
  7. Check with n = 3: 9 + 6 = 15.

Answer: n² + 2n

When the second difference is not 2, the n² number changes. A second difference of 4 means 4 ÷ 2 = 2, so the nth term starts with 2n².

Worked example

Find the nth term of 3, 10, 21, 36.

  1. The first differences are 7, 11, 15. The second differences are 4, 4. The sequence is quadratic.
  2. Start with 2n². Its number is half of the second difference: 4 ÷ 2 = 2.
  3. The values of 2n² are 2, 8, 18, 32.
  4. Take them from the sequence: 3 − 2 = 1, 10 − 8 = 2, 21 − 18 = 3 and 36 − 32 = 4.
  5. The results 1, 2, 3, 4 are the numbers n.
  6. So the nth term is 2n² + n.
  7. Check with n = 4: 2 × 16 + 4 = 36.

Answer: 2n² + n

See it three ways

With things

Write the sequence 2, 6, 12, 20, and find the next term by adding on the differences.

As a picture

The terms 2, 6, 12, 20 and 30 with their first differences +4, +6, +8 and +10, and the second differences +2, +2 and +2 inside a dashed frame.26122030+4+6+8+10+2+2+2

In words

The differences go up by 2 each time: 4, 6, 8, 10. So the next difference is 10, and the next term is 20 + 10 = 30.

In symbols

The nth term is n² + n. For n = 5, 5² + 5 = 25 + 5 = 30.

Try it

Make a quadratic sequence

You need: paper and a pencil.

  1. Take the rule n² + 3n. Work it out for n = 1, 2, 3, 4 and 5.
  2. Write the five terms in a row. Find the first differences, then the second differences.
  3. Compare the second difference with the n² number in the rule.

What you will see: The terms are 4, 10, 18, 28, 40. The second differences are all 2, which is twice the number in front of n². The rule and the differences agree.

If you cannot try it, this is what happens.
The terms 4, 10, 18, 28 and 40 with their first differences +6, +8, +10 and +12, and the second differences +2, +2 and +2 inside a dashed frame.410182840+6+8+10+12+2+2+2

A wrong idea many people have

The second difference is the number in front of n². For 2, 6, 12, 20 the nth term starts with 2n².

Why it feels right: The second difference, 2, stands out in the table, so it seems to be the number in front of n².

The test that breaks it: Put n = 2 into 2n². It gives 2 × 4 = 8, but the second term of the sequence is 6. So 2n² is too big.

What is true: The sequence n² has second difference 2. So a n² has second difference 2a. Halve the second difference to find a. Here a = 1.

A wrong idea many people have

If the first differences are not all the same, the sequence has no rule.

Why it feels right: For 3, 7, 11, 15 the first differences are all the same, so a sequence where they change looks like a mess.

The test that breaks it: The sequence 2, 6, 12, 20 has first differences 4, 6, 8, which change. But its second differences are 2, 2, and the rule n² + n gives every term.

What is true: Take differences again. When the second differences are all the same and not 0, there is a quadratic rule.

Key words

first differences
what you get by taking each term of a sequence from the next one.
second differences
the differences of the first differences.
quadratic sequence
a sequence whose second differences are all the same and not 0, so that its nth term has an n².

Back to your guess

The answer is C: Yes. The differences of the differences are all the same and not 0, and that gives the rule.

The differences of 2, 6, 12, 20 are 4, 6, 8. Take differences again and you get 2, 2. When these second differences are all the same and not 0, the rule is a quadratic. There is a method to find it. Here the nth term is n² + n.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The picture shows the sequence 3, 7, 13, 21 and its first differences. What are its second differences?

    The terms 3, 7, 13 and 21 with their first differences +4, +6 and +8 under them.371321+4+6+8
    Tap your answer.

    1. Not this one. These are the first differences, already in the picture. The second differences are the differences of these numbers. Go to M4.9.7. Try another.

    2. Not this one. These are the last two first differences. The second differences are 6 − 4 and 8 − 6. Go to M4.9.7. Try another.

    3. Yes, that is right. The first differences are 4, 6, 8. Then 6 − 4 = 2 and 8 − 6 = 2. The second differences are 2, 2.

    See the answer

    2, 2

    The first differences are 4, 6, 8. Then 6 − 4 = 2 and 8 − 6 = 2. The second differences are 2, 2.

    • If you chose A: These are the first differences, already in the picture. The second differences are the differences of these numbers. Go to M4.9.7.
    • If you chose B: These are the last two first differences. The second differences are 6 − 4 and 8 − 6. Go to M4.9.7.

  2. Which of these sequences is quadratic?

    Tap your answer.

    1. Yes, that is right. The first differences are 3, 5, 7, 9. The second differences are 2, 2, 2: all the same, and not 0, so the sequence is quadratic.

    2. Not this one. The first differences are all 3, so it goes up in equal steps. That is a straight line sequence, not a quadratic one. Go to M4.9.7. Try another.

    3. Not this one. The first differences are 1, 2, 4, 8, and the second differences are 1, 2, 4. They are not all the same. Go to M4.9.7. Try another.

    4. Not this one. The second differences are 12, 18, 24. They are not all the same, so the cubes are not a quadratic sequence. Go to M4.9.7. Try another.

    See the answer

    2, 5, 10, 17, 26

    The first differences are 3, 5, 7, 9. The second differences are 2, 2, 2: all the same, and not 0, so the sequence is quadratic.

    • If you chose B: The first differences are all 3, so it goes up in equal steps. That is a straight line sequence, not a quadratic one. Go to M4.9.7.
    • If you chose C: The first differences are 1, 2, 4, 8, and the second differences are 1, 2, 4. They are not all the same. Go to M4.9.7.
    • If you chose D: The second differences are 12, 18, 24. They are not all the same, so the cubes are not a quadratic sequence. Go to M4.9.7.

  3. Find the nth term of 4, 7, 12, 19.

    Tap your answer.

    1. Not this one. The second difference is 2, so the n² number is half of it, which is 1, not 2. Go to M4.9.7. Try another.

    2. Not this one. After taking n² from the sequence the numbers left are 3, 3, 3, 3. A constant 3 is +3, not 3n. Go to M4.9.7. Try another.

    3. Not this one. This is a straight line rule. The first differences are 3, 5, 7, which are not all the same. Go to M4.9.7. Try another.

    4. Yes, that is right. The first differences are 3, 5, 7 and the second are 2, 2, so start with n². Taking n² from the terms leaves 3, 3, 3, 3, which is +3.

    See the answer

    n² + 3

    The first differences are 3, 5, 7 and the second are 2, 2, so start with n². Taking n² from the terms leaves 3, 3, 3, 3, which is +3.

    • If you chose A: The second difference is 2, so the n² number is half of it, which is 1, not 2. Go to M4.9.7.
    • If you chose B: After taking n² from the sequence the numbers left are 3, 3, 3, 3. A constant 3 is +3, not 3n. Go to M4.9.7.
    • If you chose C: This is a straight line rule. The first differences are 3, 5, 7, which are not all the same. Go to M4.9.7.

  4. Find the nth term of 6, 14, 26, 42.

    Tap your answer.

    1. Not this one. The second difference is 4, and the n² number is half of it: 4 ÷ 2 = 2, not 4. Go to M4.9.7. Try another.

    2. Yes, that is right. The first differences are 8, 12, 16 and the second are 4, 4. So start with 2n²: 2, 8, 18, 32. Taking these away leaves 4, 6, 8, 10, which is 2n + 2.

    3. Not this one. After taking 2n² from the terms, the numbers left are 4, 6, 8, 10. They go up in 2s from 4, so that is 2n + 2, not 4n. Go to M4.9.7. Try another.

    See the answer

    2n² + 2n + 2

    The first differences are 8, 12, 16 and the second are 4, 4. So start with 2n²: 2, 8, 18, 32. Taking these away leaves 4, 6, 8, 10, which is 2n + 2.

    • If you chose A: The second difference is 4, and the n² number is half of it: 4 ÷ 2 = 2, not 4. Go to M4.9.7.
    • If you chose C: After taking 2n² from the terms, the numbers left are 4, 6, 8, 10. They go up in 2s from 4, so that is 2n + 2, not 4n. Go to M4.9.7.

  5. What is the 10th term of the sequence with nth term n² + 3n?

    Tap your answer.

    1. Yes, that is right. Put n = 10 in: 10² + 3 × 10 = 100 + 30 = 130.

    2. Not this one. The 3n part is 3 × 10 = 30, not 3. Put n = 10 into both parts: 10 × 10 + 3 × 10. Go to M4.9.7. Try another.

    3. Not this one. This is 4 × 10. Put n = 10 into n² + 3n: 10 × 10 + 3 × 10. Go to M4.9.7. Try another.

    See the answer

    130

    Put n = 10 in: 10² + 3 × 10 = 100 + 30 = 130.

    • If you chose B: The 3n part is 3 × 10 = 30, not 3. Put n = 10 into both parts: 10 × 10 + 3 × 10. Go to M4.9.7.
    • If you chose C: This is 4 × 10. Put n = 10 into n² + 3n: 10 × 10 + 3 × 10. Go to M4.9.7.

  6. Ravi works out that the second differences of 2, 6, 12, 20 are 2, 2. He says the nth term starts with 2n². What is wrong?

    Tap your answer.

    1. Not this one. The first differences change, so they cannot give the n² number. The second difference does, after halving. Go to M4.9.7. Try another.

    2. Not this one. Test it: 2n² gives 2 when n = 1, but 8 when n = 2. The second term is 6, so 2n² is too big. Go to M4.9.7. Try another.

    3. Yes, that is right. The sequence n² has second difference 2, so an² has second difference 2a. The n² number is half the second difference.

    See the answer

    He must halve the second difference to find the number in front of n squared.

    The sequence n² has second difference 2, so an² has second difference 2a. The n² number is half the second difference.

    • If you chose A: The first differences change, so they cannot give the n² number. The second difference does, after halving. Go to M4.9.7.
    • If you chose B: Test it: 2n² gives 2 when n = 1, but 8 when n = 2. The second term is 6, so 2n² is too big. Go to M4.9.7.

  7. From an earlier chapter

    A parabola crosses the x axis at x = −2 and x = 6. At which value of x is its turning point?

    Tap your answer.

    1. Not this one. You added the two roots and did not halve. The turning point is halfway between them, so halve that sum. Go to M4.9.6. Try another.

    2. Not this one. This is the distance between the roots. The turning point is halfway between them, which is a value of x. Go to M4.9.6. Try another.

    3. Not this one. You multiplied the roots. The turning point is halfway between them: add them, then halve. Go to M4.9.6. Try another.

    4. Yes, that is right. The turning point is halfway between the roots. Halfway between −2 and 6 is (−2 + 6)/2 = 2.

    See the answer

    2

    The turning point is halfway between the roots. Halfway between −2 and 6 is (−2 + 6)/2 = 2.

    • If you chose A: You added the two roots and did not halve. The turning point is halfway between them, so halve that sum. Go to M4.9.6.
    • If you chose B: This is the distance between the roots. The turning point is halfway between them, which is a value of x. Go to M4.9.6.
    • If you chose C: You multiplied the roots. The turning point is halfway between them: add them, then halve. Go to M4.9.6.

  8. A sequence has nth term n² + 2n. Find its 6th term.

    See the answer

    48

    Put n = 6 in: 6² + 2 × 6 = 36 + 12 = 48.

  9. A sequence has nth term 3n² + 2n + 1. What is its second difference?

    See the answer

    6

    The terms for n = 1, 2, 3 are 6, 17, 34. The first differences are 11 and 17, so the second difference is 6. It is 2 × 3, twice the number in front of n².

Keep it

Copy this into your exercise book.

Find the differences twice. If the second differences match and are not 0, it is quadratic. Halve the second difference for the n² number. Take n² away, then find the rest.

Draw: The terms 2, 6, 12, 20 with first differences 4, 6, 8 and second differences 2, 2.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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