Study Companion
Topic 01 | Numerical reasoning

Number series & sequences

Selective prep: continue arithmetic, geometric, mixed and alternating sequences. Find a missing term in the middle of a series.

20-30 min Printable practice Answer key Challenge included
How to use this page

Read the explanation, work through the examples, then complete the core practice before printing.

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What you will learn

Worked example 0 Diagnosing a sequence

Question: Next term of 3, 7, 11, 15, ?   (a) 18 (b) 19 (c) 20 (d) 23

Step 1. Gaps: 7-3 = 4, 11-7 = 4, 15-11 = 4. All equal. Arithmetic with gap +4.

Step 2. Next term = 15 + 4 = 19. Answer: (b).

Step 3. Distractor check: 18 = 15 + 3 (off-by-one on the gap). 20 is a “near” answer. 23 = 15 + 8 (doubled the gap). Students who rush commonly pick 18 or 23.

1. Arithmetic sequences (constant gap)

Same amount added each step. The gap can be positive or negative.

Worked example 1 Decreasing arithmetic

40, 33, 26, 19, ?

Gap: 33 - 40 = -7. 26 - 33 = -7. 19 - 26 = -7. Next = 19 - 7 = 12.

2. Geometric sequences (constant ratio)

Each term is the previous term multiplied by a fixed number.

Worked example 2 Doubling

2, 4, 8, 16, ? (a) 20 (b) 24 (c) 32 (d) 48

Ratio: 4/2 = 2, 8/4 = 2, 16/8 = 2. Each term doubles. Next = 16 * 2 = 32. Answer: (c).

Common trap: 16 + 16 = 32 happens to be correct here, which can trick students into using “add previous” on a different question where it wouldn’t work. Always verify the ratio on all three pairs.

3. Growing-gap sequences (arithmetic of differences)

The gaps themselves are an arithmetic sequence. Differences of differences (the second differences) are constant.

Worked example 3 Gaps growing by 1

1, 2, 4, 7, 11, ?

Gaps: 1, 2, 3, 4, so next gap is 5. Next term = 11 + 5 = 16.

Equivalently, this is “add the next counting number”: +1, +2, +3, +4, +5.

4. Interleaved / alternating sequences

Two independent sequences take turns. Split them out.

Worked example 4 Two patterns at once

1, 10, 3, 20, 5, 30, 7, ?

Odd positions (1st, 3rd, 5th, 7th): 1, 3, 5, 7 — odd numbers, gap +2.

Even positions (2nd, 4th, 6th, 8th): 10, 20, 30, ? — multiples of 10, gap +10.

The 8th term is an even position, so it is 40.

Drill pack

10 minutes. Write the gap under the sequence before you look at the options.

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Target: 10 minutes. Time yourself with a watch or phone timer — no pausing.
Fluency

Timed drill

    1. Next term: 5, 10, 15, 20, ? (a) 22 (b) 24 (c) 25 (d) 30
    2. Next term: 2, 4, 8, 16, 32, ? (a) 48 (b) 60 (c) 64 (d) 96
    3. Next term: 50, 44, 38, 32, ? (a) 24 (b) 26 (c) 27 (d) 28
    4. Next term: 1, 3, 6, 10, 15, ? (a) 18 (b) 20 (c) 21 (d) 24
    5. Next term: 81, 27, 9, 3, ? (a) 0 (b) 1 (c) 2 (d) 13\tfrac{1}{3}
    6. Missing term: 4, 7, ?, 13, 16 (a) 9 (b) 10 (c) 11 (d) 12
    7. Next term: 1, 4, 9, 16, 25, ? (a) 30 (b) 32 (c) 36 (d) 49
    8. Next term: 100, 50, 25, ? (a) 10 (b) 12 (c) 12.5 (d) 15
    9. Next term: 2, 5, 10, 17, 26, ? (a) 33 (b) 35 (c) 37 (d) 36
    10. Next term: 3, 6, 12, 24, ? (a) 36 (b) 42 (c) 48 (d) 60
    11. Next term: 1, 1, 2, 3, 5, 8, ? (a) 11 (b) 12 (c) 13 (d) 14
    12. Next term: 20, 17, 14, 11, ? (a) 7 (b) 8 (c) 9 (d) 10
    13. Missing term: 2, 6, 18, ?, 162 (a) 36 (b) 48 (c) 54 (d) 72
    14. Next pair: (1, 10), (2, 20), (3, 30), (4, ?) (a) 35 (b) 40 (c) 45 (d) 50
    15. Next term: 1, 8, 27, 64, ? (a) 81 (b) 100 (c) 125 (d) 128
    16. Interleaved — next term: 1, 100, 2, 90, 3, 80, 4, ? (a) 60 (b) 65 (c) 70 (d) 75
    17. Next term: 3, 7, 15, 31, ? (a) 47 (b) 55 (c) 63 (d) 62
    18. Missing term: 5, 10, 20, 40, ?, 160 (a) 60 (b) 70 (c) 80 (d) 90
    19. Next term: 100, 90, 81, 73, ? (a) 64 (b) 65 (c) 66 (d) 68
    20. Next term: 2, 3, 5, 8, 13, 21, ? (a) 28 (b) 34 (c) 32 (d) 30

Challenge

Mix of ratios, growing gaps, and Fibonacci-style patterns.

Reasoning

Harder patterns

    1. Next term: 1, 2, 6, 24, 120, ? (a) 360 (b) 480 (c) 600 (d) 720
    2. Next term: 2, 3, 5, 9, 17, ? (a) 26 (b) 31 (c) 33 (d) 34
    3. Missing term: 1, 4, 9, ?, 25, 36 (a) 13 (b) 14 (c) 16 (d) 18
    4. Next term: 1, 3, 7, 15, 31, ? (a) 47 (b) 63 (c) 55 (d) 62
    5. Next term: 2, 6, 12, 20, 30, ? (a) 40 (b) 42 (c) 44 (d) 45
    6. Next term: 64, 32, 16, 8, ?, 2 — what is the 5th term? (a) 3 (b) 4 (c) 5 (d) 6
    7. Next term: 1, 2, 4, 8, 15, 26, ? (a) 40 (b) 42 (c) 44 (d) 46
    8. Interleaved — next term: 1, 2, 2, 4, 3, 6, 4, 8, 5, ? (a) 9 (b) 10 (c) 11 (d) 12
Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Each answer names the pattern type (arithmetic / geometric / growing-gap / Fibonacci / interleaved) before the arithmetic.

Timed drill

    1. (c) 25. Arithmetic, gap +5.
    2. (c) 64. Geometric, ratio ×2\times 2. 32×2=6432 \times 2 = 64.
    3. (b) 26. Arithmetic, gap -6. 32−6=2632 - 6 = 26.
    4. (c) 21. Growing gaps +2, +3, +4, +5, next +6. 15+6=2115 + 6 = 21. (Triangular numbers.)
    5. (b) 1. Geometric, ratio ÷3\div 3. 3÷3=13 \div 3 = 1.
    6. (b) 10. Arithmetic, gap +3. 7+3=107 + 3 = 10.
    7. (c) 36. Squares: 12,22,32,42,52,621^2, 2^2, 3^2, 4^2, 5^2, 6^2. Next is 3636.
    8. (c) 12.5. Geometric, ratio ÷2\div 2. 25÷2=12.525 \div 2 = 12.5.
    9. (c) 37. Gaps +3, +5, +7, +9, next +11. 26+11=3726 + 11 = 37.
    10. (c) 48. Geometric, ratio ×2\times 2. 24×2=4824 \times 2 = 48.
    11. (c) 13. Fibonacci: add the previous two. 5+8=135 + 8 = 13.
    12. (b) 8. Arithmetic, gap -3. 11−3=811 - 3 = 8.
    13. (c) 54. Geometric, ratio ×3\times 3. 18×3=5418 \times 3 = 54 and 54×3=16254 \times 3 = 162 checks.
    14. (b) 40. Second value = first ×10\times 10. 4×10=404 \times 10 = 40.
    15. (c) 125. Cubes: 1,23,33,43,531, 2^3, 3^3, 4^3, 5^3. Next is 125125.
    16. (c) 70. Interleaved. Odd positions 1, 2, 3, 4 (+1). Even positions 100, 90, 80, ? (-10). Next even term =70= 70.
    17. (c) 63. Rule ×2+1\times 2 + 1: 3→7→15→31→633 \to 7 \to 15 \to 31 \to 63.
    18. (c) 80. Geometric, ratio ×2\times 2. 40×2=8040 \times 2 = 80 and 80×2=16080 \times 2 = 160.
    19. (c) 66. Gaps -10, -9, -8, next -7. 73−7=6673 - 7 = 66.
    20. (b) 34. Fibonacci-style: 13+21=3413 + 21 = 34.

Challenge

    1. (d) 720. Ratios grow: ×2,×3,×4,×5\times 2, \times 3, \times 4, \times 5, next ×6\times 6. 120×6=720120 \times 6 = 720. (Factorials: 1!,2!,3!,4!,5!,6!1!, 2!, 3!, 4!, 5!, 6!.)
    2. (c) 33. Rule ×2−1\times 2 - 1: 2,3,5,9,17,332, 3, 5, 9, 17, 33.
    3. (c) 16. Squares. Missing is 42=164^2 = 16.
    4. (b) 63. Rule ×2+1\times 2 + 1: 31×2+1=6331 \times 2 + 1 = 63.
    5. (b) 42. Gaps +4, +6, +8, +10, next +12. 30+12=4230 + 12 = 42. (Equivalently n(n+1)n(n+1).)
    6. (b) 4. Geometric halving. 8÷2=48 \div 2 = 4, and 4÷2=24 \div 2 = 2 checks.
    7. (b) 42. First differences 1, 2, 4, 7, 11; second differences 1, 2, 3, 4, next 5. So next first difference =11+5=16= 11 + 5 = 16, and 26+16=4226 + 16 = 42.
    8. (b) 10. Interleaved. Odd positions 1, 2, 3, 4, 5 (+1). Even positions 2, 4, 6, 8, ? (+2). Next even term =10= 10.

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