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Form 5 Sequences Practice

Form 5 · 80 practice questions with instant marking and worked solutions.

Sequences is an important part of the Form 5 mathematics curriculum. This topic develops terms, indices, general terms, arithmetic and geometric structure. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Sequences has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 80 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Sequences; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Sequences set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Sequences should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 80. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    In Sequences, calculate $a_1=7,\ d=3,\ n=5$. Choose the correct answer.
  2. 2 Basic
    In Sequences, calculate $a_1=8,\ d=4,\ n=6$. Give the numerical value only.

  3. 3 Basic
    In Sequences, calculate $a_1=9,\ d=5,\ n=7$. Show the essential working and give the numerical value.

  4. 4 Basic
    In Sequences, the value of $a_1=6,\ d=6,\ n=8$ is 49. Is this statement true?
  5. 5 Basic
    An arithmetic sequence has first term $-2$ and common difference $2$. Find the 19th term. Choose the correct answer.
  6. 6 Basic
    An arithmetic sequence has first term $6$ and common difference $2$. Find the sum of the first 16 terms. Give the answer only.

  7. 7 Basic
    A geometric sequence has first term $1$ and common ratio $3$. Find the 7th term. Show the essential working and give the answer.

  8. 8 Basic
    A geometric sequence has first term $3$ and common ratio $2$. The sum of the first 6 terms is 96. Is this true?
  9. 9 Basic
    The first three terms of an arithmetic sequence are $17,14,11$. Find the fifth term.
  10. 10 Basic
    A geometric sequence is $8,-4,2,-1,\dots$. Find the next term as a simplified fraction.

  11. 11 Basic
    Insert a number between $7$ and $19$ so that the three numbers, in that order, form an arithmetic sequence. Find the inserted number.

  12. 12 Basic
    Is this statement true: "An infinite geometric series with first term $3$ and common ratio $-2$ has a finite sum"?
  13. 13 Basic
    An arithmetic sequence has first term $13$ and common difference $-1$. Find the 21st term. Choose the correct answer.
  14. 14 Basic
    An arithmetic sequence has first term $11$ and common difference $1$. Find the sum of the first 19 terms. Give the answer only.

  15. 15 Basic
    A geometric sequence has first term $1$ and common ratio $3$. Find the 6th term. Show the essential working and give the answer.

  16. 16 Basic
    A geometric sequence has first term $3$ and common ratio $-2$. The sum of the first 6 terms is -96. Is this true?
  17. 17 Standard
    In Sequences, calculate $a_1=11,\ d=2,\ n=9$. Choose the correct answer.
  18. 18 Standard
    In Sequences, calculate $a_1=12,\ d=3,\ n=4$. Give the numerical value only.

  19. 19 Standard
    In Sequences, calculate $a_1=13,\ d=4,\ n=5$. Show the essential working and give the numerical value.

  20. 20 Standard
    In Sequences, the value of $a_1=10,\ d=5,\ n=6$ is 35. Is this statement true?
  21. 21 Standard
    An infinite geometric series has first term $3$ and common ratio $\frac{3}{4}$. Find the sum to infinity. Choose the correct answer.
  22. 22 Standard
    The $n$th term of the arithmetic sequence $0, 5, 10, \dots$ is $145$. Find the value of $n$. Give the answer only.

  23. 23 Standard
    The 4th term of an arithmetic sequence is $3$ and the 7th term is $6$. Find the common difference. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    An arithmetic sequence has first term $8$ and fifth term $24$. Answer each part; give the answer only.
    1. (a) Find the common difference.
    2. (b) Find the 11th term.
    3. (c) Find the sum of the first 11 terms.
  25. 25 Standard
    Three positive numbers $4,b,36$, in that order, form a geometric sequence. Find $b$.
  26. 26 Standard
    The third and eighth terms of an arithmetic sequence are $11$ and $26$. Find the first term.

  27. 27 Standard
    The sum of the first $n$ terms of a sequence is $S_n=3n^2+2n$. Find the fifth term.

  28. 28 Standard DSE long question
    Ming saves HK$20 on the first day and HK$5 more than the previous day on each subsequent day.
    1. (a) Find the amount saved on day $10$, in dollars.
    2. (b) Find the total saved in the first $10$ days, in dollars.
    3. (c) Find the amount saved on day $18$, in dollars.
  29. 29 Standard
    An infinite geometric series has first term $8$ and common ratio $-\frac{1}{3}$. Find the sum to infinity. Choose the correct answer.
  30. 30 Standard
    The $n$th term of the arithmetic sequence $10, 17, 24, \dots$ is $122$. Find the value of $n$. Give the answer only.

  31. 31 Standard
    The 3rd term of an arithmetic sequence is $1$ and the 8th term is $-9$. Find the common difference. Show the essential working and give the answer.

  32. 32 Standard DSE long question
    An arithmetic sequence has first term $11$ and fifth term $19$. Answer each part; give the answer only.
    1. (a) Find the common difference.
    2. (b) Find the 27th term.
    3. (c) Find the sum of the first 27 terms.
  33. 33 Advanced
    In Sequences, calculate $a_1=15,\ d=6,\ n=7$. Choose the correct answer.
  34. 34 Advanced
    In Sequences, calculate $a_1=16,\ d=2,\ n=8$. Give the numerical value only.

  35. 35 Advanced
    In Sequences, calculate $a_1=17,\ d=3,\ n=9$. Show the essential working and give the numerical value.

  36. 36 Advanced
    In Sequences, the value of $a_1=14,\ d=4,\ n=4$ is 27. Is this statement true?
  37. 37 Advanced
    An arithmetic sequence has first term $-2$ and common difference $-6$. Find the 31st term. Choose the correct answer.
  38. 38 Advanced
    An arithmetic sequence has first term $4$ and common difference $5$. Find the sum of the first 23 terms. Give the answer only.

  39. 39 Advanced
    A geometric sequence has first term $4$ and common ratio $\frac{1}{2}$. The 7th term is 1/16. Is this true?
  40. 40 Advanced DSE long question
    A geometric sequence has first term $16$ and common ratio $\frac{1}{3}$. Answer each part; give the answer only.
    1. (a) Find the 4th term (as a fraction in lowest terms).
    2. (b) Find the sum of the first 4 terms (as a fraction in lowest terms).
    3. (c) Find the sum to infinity.
  41. 41 Advanced
    An arithmetic sequence has first term $12$ and common difference $-2$. Its first $n$ terms sum to zero, where $n$ is a positive integer. Find $n$.
  42. 42 Advanced
    The second and fifth terms of a geometric sequence are $6$ and $162$, and the common ratio is positive. Find the first term.

  43. 43 Advanced
    Is this statement true: "The tenth term of the arithmetic sequence $3,7,11,\dots$ is $39$"?
  44. 44 Advanced DSE long question
    A convergent infinite geometric series has first term $5$ and sum $20$.
    1. (a) Find the common ratio as a simplified fraction.
    2. (b) Find the fourth term as a simplified fraction.
    3. (c) Find the sum of the first three terms as a simplified fraction.
  45. 45 Advanced
    An arithmetic sequence has first term $-5$ and common difference $-3$. Find the 12th term. Choose the correct answer.
  46. 46 Advanced
    An arithmetic sequence has first term $12$ and common difference $3$. Find the sum of the first 6 terms. Give the answer only.

  47. 47 Advanced
    A geometric sequence has first term $4$ and common ratio $2$. The 4th term is 32. Is this true?
  48. 48 Advanced DSE long question
    A geometric sequence has first term $48$ and common ratio $\frac{1}{2}$. Answer each part; give the answer only.
    1. (a) Find the 4th term.
    2. (b) Find the sum of the first 4 terms.
    3. (c) Find the sum to infinity.
  49. 49 Challenging
    In Sequences, calculate $a_1=19,\ d=5,\ n=5$. Choose the correct answer.
  50. 50 Challenging
    In Sequences, calculate $a_1=20,\ d=6,\ n=6$. Give the numerical value only.

  51. 51 Challenging
    In Sequences, calculate $a_1=21,\ d=2,\ n=7$. Show the essential working and give the numerical value.

  52. 52 Challenging
    In Sequences, the value of $a_1=18,\ d=3,\ n=8$ is 39. Is this statement true?
  53. 53 Challenging
    A geometric sequence has first term $1$ and common ratio $2$. Find the sum of the first 6 terms. Choose the correct answer.
  54. 54 Challenging
    An infinite geometric series has first term $3$ and common ratio $-\frac{1}{2}$. Find the sum to infinity. Show the essential working and give the answer.

  55. 55 Challenging
    The $n$th term of the arithmetic sequence $8, 12, 16, \dots$ is $144$. The value of $n$ is 34. Is this true?
  56. 56 Challenging DSE long question
    An arithmetic sequence has first term $-6$ and fifth term $2$. Answer each part; give the answer only.
    1. (a) Find the common difference.
    2. (b) Find the 19th term.
    3. (c) Find the sum of the first 19 terms.
  57. 57 Challenging
    The numbers $x-1,2x+3,4x+1$, in that order, form an arithmetic sequence. Find $x$.
  58. 58 Challenging
    A convergent infinite geometric series has sum $24$ and second term $6$, with $0

  59. 59 Challenging
    Is this statement true: "The sequence $a_n=n^2$ for $n=1,2,\dots$ is arithmetic"?
  60. 60 Challenging DSE long question
    An auditorium has $18$ seats in the first row and $2$ more seats in each successive row. The last row has $50$ seats.
    1. (a) Find the number of rows.
    2. (b) Find the total number of seats.
    3. (c) Find the total number of seats in the first $10$ rows.
  61. 61 Challenging
    A geometric sequence has first term $4$ and common ratio $-2$. Find the sum of the first 8 terms. Choose the correct answer.
  62. 62 Challenging
    An infinite geometric series has first term $9$ and common ratio $\frac{1}{3}$. Find the sum to infinity (as a fraction in lowest terms). Show the essential working and give the answer.

  63. 63 Challenging
    The $n$th term of the arithmetic sequence $1, 3, 5, \dots$ is $25$. The value of $n$ is 12. Is this true?
  64. 64 Challenging DSE long question
    An arithmetic sequence has first term $10$ and fifth term $26$. Answer each part; give the answer only.
    1. (a) Find the common difference.
    2. (b) Find the 22nd term.
    3. (c) Find the sum of the first 22 terms.
  65. 65 Elite
    In Sequences, calculate $a_1=23,\ d=4,\ n=9$. Choose the correct answer.
  66. 66 Elite
    In Sequences, calculate $a_1=24,\ d=5,\ n=4$. Give the numerical value only.

  67. 67 Elite
    In Sequences, calculate $a_1=25,\ d=6,\ n=5$. Show the essential working and give the numerical value.

  68. 68 Elite
    In Sequences, the value of $a_1=22,\ d=2,\ n=6$ is 33. Is this statement true?
  69. 69 Elite
    The 2nd term of an arithmetic sequence is $6$ and the 11th term is $-30$. Find the common difference. Give the answer only.

  70. 70 Elite
    An arithmetic sequence has first term $7$ and common difference $-2$. Find the 21st term. Show the essential working and give the answer.

  71. 71 Elite
    An arithmetic sequence has first term $2$ and common difference $-3$. The sum of the first 35 terms is -1715. Is this true?
  72. 72 Elite DSE long question
    A geometric sequence has first term $16$ and common ratio $-\frac{1}{2}$. Answer each part; give the answer only.
    1. (a) Find the 4th term.
    2. (b) Find the sum of the first 4 terms.
    3. (c) Find the sum to infinity (as a fraction in lowest terms).
  73. 73 Elite
    Two arithmetic sequences have $n$th terms $5n-2$ and $3n+10$. Find the positive integer $n$ for which their first $n$ terms have equal sums.

  74. 74 Elite
    The first three terms of a geometric sequence are $x-2,x,2x+3$, all positive. Find the common ratio as a simplified fraction.

  75. 75 Elite
    Is this statement true: "In a geometric series with first term $9$ and ratio $1/3$, the sum from the fourth term onwards is $1/2$"?
  76. 76 Elite DSE long question
    A convergent infinite geometric series has first term $12$. The sum from the second term onwards is $6$.
    1. (a) Find the common ratio as a simplified fraction.
    2. (b) Find the third term as a simplified fraction.
    3. (c) Find the sum from the fifth term onwards as a simplified fraction.
  77. 77 Elite
    The 5th term of an arithmetic sequence is $5$ and the 12th term is $12$. Find the common difference. Give the answer only.

  78. 78 Elite
    An arithmetic sequence has first term $7$ and common difference $4$. Find the 45th term. Show the essential working and give the answer.

  79. 79 Elite
    An arithmetic sequence has first term $6$ and common difference $4$. The sum of the first 20 terms is 880. Is this true?
  80. 80 Elite DSE long question
    A geometric sequence has first term $96$ and common ratio $\frac{1}{2}$. Answer each part; give the answer only.
    1. (a) Find the 4th term.
    2. (b) Find the sum of the first 4 terms.
    3. (c) Find the sum to infinity.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Sequences

This topic needs particular care with terms, indices, general terms, arithmetic and geometric structure. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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