← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 2 Algebra and graphs · Chapter 15 · pp. 153–165

Sequences

A sequence is a list of numbers made by a rule: 2, 4, 6, 8, … or 1, 4, 9, 16, … Once you know the rule you can write any term without listing the ones before it. This chapter gives you the tools to find that rule, from a simple “add 5 each time” to quadratic, cubic and exponential patterns.

Learning objectives

  1. Use the notation u₁, u₂, … uₙ and the term-to-term rule uₙ₊₁ = uₙ + d.
  2. Find the formula for the nth term of an arithmetic sequence.
  3. Find the nth term of a quadratic or cubic sequence with a table of differences.
  4. Work with exponential sequences and compound interest. (Extended)
  5. Spot key sequences (squares, cubes, powers of 2, triangle numbers) hiding in a pattern. (Extended)

Book exercises: 15.1 to 15.5 and Student assessments 1 and 2 (pp. 153–165).

Arithmetic sequences

🔑 Key ideas

  • u₁ is the 1st term, u₂ the 2nd, and uₙ the nth term.
  • In an arithmetic sequence the same number, the common difference d, is added every time.
  • Term-to-term rule: uₙ₊₁ = uₙ + d. You must also give one term, for example u₁ = 7, or the sequence is not fixed.
  • Formula for the nth term: uₙ = dn + b. The coefficient of n is d; then b is whatever makes the 1st term right.
7 n = 1 +5 12 n = 2 +5 17 n = 3 +5 22 n = 4 +5 27 n = 5 uₙ = 5n + 2 (d = 5, and 5 × 1 + 2 = 7)
Every arc adds the common difference. The coefficient of n in the formula is that same number.

📘 Example 1 · uₙ₊₁ = uₙ + 3 with u₁ = 0. Find u₂, u₃ and u₄.

  1. u₂ = u₁ + 3 = 0 + 3 = 3.
  2. u₃ = u₂ + 3 = 3 + 3 = 6.
  3. u₄ = u₃ + 3 = 6 + 3 = 9. The terms are 3, 6, 9: arithmetic, with d = 3.

📘 Example 2 · Find the nth term of 12, 7, 2, −3, −8, …

  1. Subtract neighbours: 7 − 12 = −5, 2 − 7 = −5. So d = −5.
  2. The formula starts uₙ = −5n + b.
  3. At n = 1: −5 × 1 + b = 12, so b = 17.
  4. Answer: uₙ = −5n + 17.
  5. Check n = 5: −25 + 17 = −8 ✓.

📘 Example 3 · An arithmetic sequence has u₂ = 11 and u₆ = 27. Find uₙ, then u₅₀.

  1. From u₂ to u₆ is 4 steps, and the terms rise by 27 − 11 = 16.
  2. One step: d = 16 ÷ 4 = 4, so uₙ = 4n + b.
  3. At n = 2: 4 × 2 + b = 11, so b = 3. uₙ = 4n + 3.
  4. Check n = 6: 24 + 3 = 27 ✓.
  5. u₅₀ = 4 × 50 + 3 = 203.

📘 Example 4 · Is 99 a term of uₙ = 4n + 3? Is 100?

  1. Set 4n + 3 = 99, so 4n = 96 and n = 24.
  2. n is a whole number, so 99 is the 24th term.
  3. Set 4n + 3 = 100, so 4n = 97 and n = 24.25.
  4. A position must be a whole number, so 100 is not a term.

✏️ Try it

  1. uₙ₊₁ = uₙ − 4 and u₁ = 20. Find u₂, u₃ and u₄.
  2. Find the nth term of 8, 11, 14, 17, … and its 10th term.
  3. Is 60 a term of uₙ = 5n − 3?
Answers

1. 16, 12, 8 · 2. uₙ = 3n + 5; u₁₀ = 35 · 3. 5n − 3 = 60 gives n = 12.6, so no

Next: quadratic and cubic rules →

Quadratic and cubic rules: the method of differences

🔑 Key ideas

  • Subtract neighbouring terms to get the 1st differences. Subtract those to get the 2nd differences, and so on.
  • The first row that is constant tells you the type of rule.
  • Linear: 1st constant, uₙ = an + b with a = the difference.
  • Quadratic: 2nd constant, uₙ = an² + bn + c with 2a = the 2nd difference.
  • Cubic: 3rd constant, uₙ = an³ + bn² + cn + d with 6a = the 3rd difference.
Position Term 1st difference 2nd difference 1 0 2 7 3 18 4 33 5 52 7 11 15 19 4 4 4 2nd differences constant (4), so quadratic. 2a = 4, a = 2.
The sequence 0, 7, 18, 33, 52. Each row is the gaps of the row above it.
1st differences constant? No Yes Linear an + b 2nd differences constant? No Yes Quadratic an² + bn + c 3rd differences constant? Yes Cubic an³ + bn² + cn + d None constant? Look for a common ratio or a key sequence.
Take differences until a row is constant. That row number is the highest power of n.

📘 Example 5 · Quadratic: find the nth term of 0, 7, 18, 33, 52, …

  1. 1st differences: 7, 11, 15, 19. Not constant.
  2. 2nd differences: 4, 4, 4. Constant, so uₙ = an² + bn + c.
  3. 2a = 4, so a = 2.
  4. The first 1st difference is 3a + b = 7, so 6 + b = 7 and b = 1.
  5. Term 1 is a + b + c = 0, so 2 + 1 + c = 0 and c = −3.
  6. Answer: uₙ = 2n² + n − 3.
  7. Check n = 4: 32 + 4 − 3 = 33 ✓.

📘 Example 6 · Quadratic: find the nth term of 5, 9, 15, 23, 33, …

  1. 1st differences: 4, 6, 8, 10. 2nd differences: 2, 2, 2.
  2. 2a = 2, so a = 1.
  3. 3a + b = 4, so 3 + b = 4 and b = 1.
  4. a + b + c = 5, so 1 + 1 + c = 5 and c = 3.
  5. Answer: uₙ = n² + n + 3.
  6. Check n = 5: 25 + 5 + 3 = 33 ✓.

📘 Example 7 · Cubic: find the nth term of −6, −8, −6, 6, 34, …

  1. 1st differences: −2, 2, 12, 28.
  2. 2nd differences: 4, 10, 16. Still not constant.
  3. 3rd differences: 6, 6. Constant, so uₙ = an³ + bn² + cn + d.
  4. 6a = 6, so a = 1.
  5. The first 2nd difference is 12a + 2b = 4: 12 + 2b = 4, so b = −4.
  6. The first 1st difference is 7a + 3b + c = −2: 7 − 12 + c = −2, so c = 3.
  7. Term 1 is a + b + c + d = −6: 1 − 4 + 3 + d = −6, so d = −6.
  8. Answer: uₙ = n³ − 4n² + 3n − 6.
  9. Check n = 4: 64 − 64 + 12 − 6 = 6 ✓.

🧪 Try it: nth-term finder

Type at least four terms (five for a cubic), separated by commas. Fractions such as 1/2 work. The tool finds the rule step by step.

    ✏️ Try it

    1. Find the nth term of 3, 6, 11, 18, 27, …
    2. A sequence has all 2nd differences equal to 10. What is the coefficient of n²?
    3. Find the nth term of 4, 11, 30, 67, 128, …
    Answers

    1. 1st: 3, 5, 7, 9; 2nd: 2, so a = 1, b = 0, c = 2: uₙ = n² + 2 · 2. 2a = 10, so 5 · 3. 1st: 7, 19, 37, 61; 2nd: 12, 18, 24; 3rd: 6, so a = 1, b = 0, c = 0, d = 3: uₙ = n³ + 3

    Next: exponential sequences →

    Exponential sequences Extended

    🔑 Key ideas

    • An exponential (or geometric) sequence multiplies by the same common ratio r each time.
    • Find r by dividing a term by the one before it.
    • Term-to-term rule: uₙ₊₁ = r × uₙ. nth term: uₙ = a × rⁿ⁻¹, where a is the 1st term.
    • The power is n − 1, not n, because the 1st term has not been multiplied yet.
    3 × 2 6 × 2 12 × 2 24 × 2 48 r = 2: uₙ = 3 × 2ⁿ⁻¹ 27 × ⅓ 9 × ⅓ 3 × ⅓ 1 × ⅓ ⅓ r = ⅓: uₙ = 27 × (⅓)ⁿ⁻¹
    Top: growing, r = 2. Bottom: shrinking, r = ⅓. A ratio between 0 and 1 gives smaller and smaller terms.

    📘 Example 8 · 3, 6, 12, 24, 48, … Find r, uₙ and u₁₀.

    1. Divide: 6 ÷ 3 = 2 and 12 ÷ 6 = 2. So r = 2.
    2. a = 3, so uₙ = 3 × 2ⁿ⁻¹.
    3. u₁₀ = 3 × 2⁹ = 3 × 512 = 1536.

    📘 Example 9 · An exponential sequence has u₂ = 6 and u₅ = 162. Find r and u₁.

    1. From u₂ to u₅ is 3 steps, so r³ = 162 ÷ 6 = 27.
    2. r = ∛27 = 3.
    3. Go back one step: u₁ = 6 ÷ 3 = 2. So uₙ = 2 × 3ⁿ⁻¹.
    4. Check u₅ = 2 × 3⁴ = 2 × 81 = 162 ✓.

    Compound interest is an exponential sequence

    Interest added to the balance earns interest too. A 10% rise has multiplier 1.1, so the balance is multiplied by 1.1 every year.

    $100.00 Year 0 $110.00 Year 1 $121.00 Year 2 $133.10 Year 3 $146.41 Year 4 $161.05 Year 5
    $100 at 10% a year: × 1.1 each year, so 100 × 1.1⁵ = $161.05 after 5 years. The bars grow faster each year.

    📘 Example 10 · $1500 at 6% compound interest for 10 years. How much interest?

    1. 6% gives multiplier 1.06.
    2. Balance after 10 years: 1500 × 1.06¹⁰ = 2686.27 (to 2 d.p.).
    3. Interest = 2686.27 − 1500 = $1186.27.

    📘 Example 11 · A car worth $8000 loses 15% of its value each year. Value after 3 years?

    1. Losing 15% leaves 85%, so the multiplier is 0.85.
    2. Value = 8000 × 0.85³ = 8000 × 0.614125.
    3. Answer: $4913. Because you keep multiplying by 0.85, the value gets small but never reaches 0.

    ✏️ Try it

    1. For 5, 10, 20, 40, … find r and u₇.
    2. For 64, 16, 4, 1, … find r and the next two terms.
    3. $500 earns 4% compound interest a year. What is it worth after 3 years?
    Answers

    1. r = 2; u₇ = 5 × 2⁶ = 320 · 2. r = ¼; next terms ¼ and 1/16 · 3. 500 × 1.04³ = 500 × 1.124864 = $562.43

    Next: combinations of sequences →

    Combinations of sequences Extended

    Many sequences are a famous one with a small change. Spotting the family can be much quicker than a table of differences.

    Key sequenceTermsnth term
    Square numbers1, 4, 9, 16, 25, …n²
    Cube numbers1, 8, 27, 64, 125, …n³
    Powers of two2, 4, 8, 16, 32, …2ⁿ
    Triangle numbers1, 3, 6, 10, 15, …½n(n + 1)
    Squares n² Your sequence 1 +1 2 4 +1 5 9 +1 10 16 +1 17 25 +1 26 Every term is 1 more than a square, so uₙ = n² + 1
    Compare with a key sequence and describe the difference. 2, 5, 10, 17, 26 is the squares plus 1.

    📘 Example 12 · 2, 8, 18, 32, … Find uₙ and the next two terms.

    1. Compare with the squares 1, 4, 9, 16. Every term is double.
    2. So uₙ = 2n².
    3. n = 5: 2 × 25 = 50. n = 6: 2 × 36 = 72. Next terms: 50, 72.

    📘 Example 13 · 1, 3, 7, 15, 31, … Find uₙ and the next two terms.

    1. Compare with the powers of two 2, 4, 8, 16, 32. Every term is 1 less.
    2. So uₙ = 2ⁿ − 1.
    3. n = 6: 64 − 1 = 63. n = 7: 128 − 1 = 127. Next terms: 63, 127.

    📘 Example 14 · 11, 13, 16, 20, … Find uₙ and the next two terms.

    1. Compare with the triangle numbers 1, 3, 6, 10. Each term is 10 more.
    2. So uₙ = ½n(n + 1) + 10.
    3. n = 5: 15 + 10 = 25. n = 6: 21 + 10 = 31. Next terms: 25, 31.

    ✏️ Try it

    1. 7, 14, 28, 56, …: find uₙ and the next two terms.
    2. 0, 3, 8, 15, …: find uₙ and the next two terms.
    3. 10, 13, 18, 25, …: find uₙ and the next two terms.
    Answers

    1. Doubling from 7: uₙ = 7 × 2ⁿ⁻¹; next 112, 224 · 2. One less than the squares: uₙ = n² − 1; next 24, 35 · 3. Nine more than the squares: uₙ = n² + 9; next 34, 45

    Next: summary and quiz →

    Chapter summary

    • uₙ is the nth term. Arithmetic: uₙ₊₁ = uₙ + d and uₙ = dn + b.
    • Method of differences: the first constant row is 1st (linear), 2nd (quadratic: 2a = difference) or 3rd (cubic: 6a = difference).
    • Substitute n = 1, 2, 3 into the general rule and match each row of the table to find the coefficients.
    • Exponential: uₙ₊₁ = r × uₙ and uₙ = a × rⁿ⁻¹. Compound interest multiplies by 1 + rate each year.
    • Key sequences: n², n³, 2ⁿ and ½n(n + 1). Compare first, then adjust.
    • A number is a term only if its position n comes out as a whole number.

    📄 Practice questions

    The book's Exercises 15.1 to 15.5 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. uₙ₊₁ = uₙ + 6 and u₁ = 4. What is u₃?
    2. What is the common difference of 11, 8, 5, 2, …?
    3. Find the nth term of 5, 9, 13, 17, …
    4. uₙ = 3n − 7. What is the 20th term?
    5. Is 75 a term of uₙ = 4n + 2?
    6. The 2nd differences of a sequence are all 8. What is the n² coefficient?
    7. Which row of differences is constant for a cubic sequence?
    8. Find the nth term of 2, 8, 18, 32, 50, …
    9. What is the common ratio of 81, 27, 9, 3, …?
    10. An exponential sequence has u₁ = 5 and r = 2. What is u₄?
    11. $400 earns 5% compound interest a year. What is it worth after 2 years?
    12. Find the nth term of 3, 5, 9, 17, 33, …

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 15 (Sequences). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.