Sequences

50 maths questions with worked answers with answers for A23–A25 Sequences in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

1. Question 1 · Term-to-term · A23

   The sequence 4, 9, 14, 19, … continues. Write the next two terms.

   Answer:

   The rule is add 5.
   24 and 29.

2. Question 2 · Term-to-term · A23

   The first term is 3 and the rule is "multiply by 2 then subtract 1". Find the first four terms.

   Answer:

   3, 3 × 2 − 1 = 5, 5 × 2 − 1 = 9, 9 × 2 − 1 = 17.
   3, 5, 9, 17.

3. Question 3 · Position-to-term · A23

   The nth term is 6n − 2. Find the 10th term.

   Answer:

   6 × 10 − 2.
   58.

4. Question 4 · Position-to-term · A23

   The nth term is n² + 3. Find the first three terms.

   Answer:

   1 + 3, 4 + 3, 9 + 3.
   4, 7, 12.

5. Question 5 · Position-to-term · A23

   The nth term is 20 − 3n. Find the first three terms.

   Answer:

   20 − 3, 20 − 6, 20 − 9.
   17, 14, 11.

6. Question 6 · Special sequences · A24

   Write the first five square numbers.

   Answer:

   1², 2², 3², 4², 5².
   1, 4, 9, 16, 25.

7. Question 7 · Special sequences · A24

   Write the first five cube numbers.

   Answer:

   1³, 2³, 3³, 4³, 5³.
   1, 8, 27, 64, 125.

8. Question 8 · Special sequences · A24

   Write the first five triangular numbers.

   Answer:

   Add 1, 2, 3, 4, 5 in turn.
   1, 3, 6, 10, 15.

9. Question 9 · Fibonacci · A24

   Write the next two terms of 1, 1, 2, 3, 5, 8, ….

   Answer:

   Each term is the sum of the previous two.
   13 and 21.

10. Question 10 · Fibonacci-type · A24

    A Fibonacci-type sequence starts 2, 5, … Find the next three terms.

    Answer:

    2 + 5 = 7, 5 + 7 = 12, 7 + 12 = 19.
    7, 12, 19.

11. Question 11 · Geometric · A24

    What is the common ratio of 3, 12, 48, 192, …?

    Answer:

    12 ÷ 3 = 4.
    The common ratio is 4.

12. Question 12 · Geometric · A24

    Write the next term of 80, 40, 20, 10, ….

    Answer:

    Each term is halved.
    5.

13. Question 13 · Geometric · A24 · Higher only

    A geometric sequence has first term 5 and common ratio 3. Find the 4th term.

    Answer:

    5 × 3³.
    135.

14. Question 14 · Geometric · A24 · Higher only

    Find the next term of 1, √2, 2, 2√2, ….

    Answer:

    The common ratio is √2.
    2√2 × √2 = 4.

15. Question 15 · Quadratic sequences · A24 · Higher only

    Is 3, 6, 11, 18, 27 linear or quadratic? Explain.

    Answer:

    First differences: 3, 5, 7, 9. Second differences: 2, 2, 2.
    The second differences are constant, so it is quadratic.

16. Question 16 · Geometric · A24 · Higher only

    Find the nth term of 2, 6, 18, 54, ….

    Answer:

    First term 2, ratio 3.
    2 × 3ⁿ⁻¹.

17. Question 17 · nth term · A25

    Find the nth term of 7, 11, 15, 19, ….

    Answer:

    The difference is 4, so start with 4n. 4n gives 4, 8, 12…, so add 3.
    4n + 3.

18. Question 18 · nth term · A25

    Find the nth term of 2, 7, 12, 17, ….

    Answer:

    The difference is 5. 5n gives 5, 10, 15…, so subtract 3.
    5n − 3.

19. Question 19 · nth term · A25

    Find the nth term of 30, 26, 22, 18, ….

    Answer:

    The difference is −4. −4n gives −4, −8…, so add 34.
    34 − 4n.

20. Question 20 · nth term · A25

    Is 75 a term of the sequence with nth term 4n + 3? Explain.

    Answer:

    4n + 3 = 75 gives 4n = 72.
    n = 18, a whole number, so yes, it is the 18th term.

21. Question 21 · nth term · A25

    Is 100 a term of the sequence 3n + 2? Explain.

    Answer:

    3n + 2 = 100 gives 3n = 98, n = 32.67.
    n is not a whole number, so no.

22. Question 22 · nth term · A25

    Find the first term of 5n + 2 that is greater than 60.

    Answer:

    5n + 2 > 60, so n > 11.6. Take n = 12.
    5 × 12 + 2 = 62.

23. Question 23 · Quadratic nth term · A25 · Higher only

    Find the nth term of 2, 5, 10, 17, 26, ….

    Answer:

    Second difference 2, so start with n². n² gives 1, 4, 9, 16; remainder is 1 each time.
    n² + 1.

24. Question 24 · Quadratic nth term · A25 · Higher only

    Find the nth term of 4, 10, 18, 28, 40, ….

    Answer:

    Second difference 2, so n². Subtract n²: 3, 6, 9, 12, 15 = 3n.
    n² + 3n.

25. Question 25 · Quadratic nth term · A25 · Higher only

    Find the nth term of 3, 9, 19, 33, 51, ….

    Answer:

    Second difference 4, so 2n². Subtract: 3 − 2, 9 − 8, 19 − 18… = 1, 1, 1.
    2n² + 1.

26. Question 26 · Quadratic nth term · A25 · Higher only

    Find the nth term of 0, 3, 8, 15, 24, ….

    Answer:

    Second difference 2, so n². Subtract n²: −1 each time.
    n² − 1.

27. Question 27 · nth term · A25

    Write a sequence whose nth term is 10n.

    Answer:

    Substitute n = 1, 2, 3, 4.
    10, 20, 30, 40, … (the multiples of 10).

28. Question 28 · Patterns · A23

    Pattern 1 has 5 sticks, pattern 2 has 9 and pattern 3 has 13. How many sticks are in pattern 10?

    Answer:

    nth term = 4n + 1.
    4 × 10 + 1 = 41.

29. Question 29 · Patterns · A23

    In the pattern 4n + 1, which pattern uses 81 sticks?

    Answer:

    4n + 1 = 81.
    n = 20.

30. Question 30 · Arithmetic · A24

    Explain what makes a sequence arithmetic.

    Answer:

    The terms go up or down by the same amount.
    It has a common difference.

31. Question 31 · Arithmetic · A24

    Find the common difference of 2.5, 3.2, 3.9, 4.6.

    Answer:

    3.2 − 2.5.
    0.7.

32. Question 32 · Term-to-term · A23

    Find the missing term: 7, __, 19, 25.

    Answer:

    The difference is 6.
    13.

33. Question 33 · Geometric · A24 · Higher only

    The 1st term of a geometric sequence is 2 and the 3rd term is 50. Find the common ratio, if it is positive.

    Answer:

    2r² = 50, so r² = 25.
    r = 5.

34. Question 34 · nth term · A25

    Find the nth term of −3, −1, 1, 3, ….

    Answer:

    The difference is 2. 2n gives 2, 4, 6…, so subtract 5.
    2n − 5.

35. Question 35 · nth term · A25

    Find the 50th term of 8, 13, 18, 23, ….

    Answer:

    nth term = 5n + 3.
    5 × 50 + 3 = 253.

36. Question 36 · Position-to-term · A23

    Use the nth term 3n² to find the 4th term.

    Answer:

    3 × 4².
    3 × 16 = 48.

37. Question 37 · Quadratic nth term · A25 · Higher only

    Find the nth term of 2, 8, 18, 32, ….

    Answer:

    Second difference 4, so 2n². Subtract 2n²: 0 each time.
    2n².

38. Question 38 · Special sequences · A24

    What do you notice about the differences between square numbers 1, 4, 9, 16, 25?

    Answer:

    The differences are 3, 5, 7, 9.
    They are consecutive odd numbers.

39. Question 39 · Special sequences · A24

    Add two consecutive triangular numbers, e.g. 6 + 10. What type of number do you get?

    Answer:

    6 + 10 = 16 and 3 + 6 = 9.
    A square number.

40. Question 40 · Term-to-term · A23

    The rule is "subtract 7". The fourth term is 5. Find the first term.

    Answer:

    Work backwards: 5 + 7 = 12, 12 + 7 = 19, 19 + 7 = 26.
    26.

41. Question 41 · nth term · A25

    Two sequences have nth terms 3n + 4 and 5n − 6. Which term is the same in both?

    Answer:

    3n + 4 = 5n − 6, so 2n = 10.
    n = 5; both have 19 as their 5th term.

42. Question 42 · Geometric · A24 · Higher only

    Find the sum of the first four terms of 1, 3, 9, 27, ….

    Answer:

    1 + 3 + 9 + 27.
    40.

43. Question 43 · Quadratic nth term · A25 · Higher only

    Find the 10th term of the sequence with nth term n² − 2n.

    Answer:

    10² − 2 × 10.
    100 − 20 = 80.

44. Question 44 · nth term · A25

    A sequence has nth term 7n − 1. Explain why every term is one less than a multiple of 7.

    Answer:

    7n is always a multiple of 7.
    So 7n − 1 is always 1 less than a multiple of 7.

45. Question 45 · Special sequences · A24

    Apart from 1, what is the smallest number that is both square and triangular?

    Answer:

    Squares: 1, 4, 9, 16, 25, 36…; triangular: 1, 3, 6, 10, 15, 21, 28, 36….
    36.

46. Question 46 · Patterns · A23

    The nth term of a pattern is 6n + 2. What do the 6 and the 2 tell you?

    Answer:

    The 6 is the common difference.
    The 2 is the difference between the term and 6n (the zero term is 2).

47. Question 47 · Quadratic nth term · A25 · Higher only

    Find the nth term of 6, 15, 28, 45, ….

    Answer:

    Second difference 4, so 2n². Subtract: 4, 7, 10, 13 = 3n + 1.
    2n² + 3n + 1.

48. Question 48 · nth term · A25

    Find the nth term of 1.5, 2, 2.5, 3, ….

    Answer:

    The difference is 0.5. 0.5n gives 0.5, 1, 1.5…, so add 1.
    0.5n + 1.

49. Question 49 · Position-to-term · A23

    The nth term is nn + 1. Find the first three terms.

    Answer:

    Substitute n = 1, 2 and 3.
    12, 23, 34.

50. Question 50 · Arithmetic · A24

    Is 2, 6, 18, 54 an arithmetic or a geometric sequence?

    Answer:

    The differences are 4, 12 and 36, which are not constant.
    It is geometric: each term is multiplied by 3.

More Edexcel GCSE Maths topics

• N1–N9 Structure and calculation
• N10–N12 Fractions, decimals and percentages
• N13–N16 Measures and accuracy
• A1–A7 Notation, vocabulary and manipulation
• A8–A16 Graphs
• A17–A22 Solving equations and inequalities
• A23–A25 Sequences
• R1–R16 Ratio, proportion and rates of change
• G1–G15 Properties and constructions
• G16–G23 Mensuration and calculation
• G24–G25 Vectors
• P1–P9 Probability
• S1–S6 Statistics

Other Maths courses: AQA GCSE Maths, OCR GCSE Maths.