Professional + Subprofessional

Civil Service Exam number sequence: four patterns cover almost every item.

Last reviewed May 2026

Number sequence is one of the listed topics under Numerical Ability, and it is among the most teachable in the section. Almost every sequence you will see falls into one of four pattern types, and once you know how to test for each, you'll solve them in under 30 seconds. Test-takers who haven't learned the pattern-recognition method guess; test-takers who have rarely miss. The CSC does not publish a per-topic breakdown, so treat any online figures about how many sequence items appear as estimates, not official counts.

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Quick facts

Primary subtest: Numerical Ability
Calculator: Not allowed
Official scope: Listed CSC numerical topic
Difficulty to improve: Low (pattern recognition)

The four pattern types: test in this order

Almost every CSC sequence is one of four types. Test them in the order below. The earlier ones are faster to check and far more common, so you rarely get past type two.

The four pattern types and how to spot each one
Type	Rule	How to test	Example
Arithmetic	Each term differs from the previous by a constant	Differences are constant	3, 7, 11, 15 (+4 each)
Geometric	Each term is the previous times a constant ratio	Ratios are constant	3, 6, 12, 24 (x2 each)
Recursive (Fibonacci-like)	Each term is built from the previous one or two	Sum of the previous two terms fits	1, 1, 2, 3, 5, 8
Second-difference	The differences themselves form a pattern	Differences of differences are constant	2, 6, 12, 20, 30

Worked detail for each type Arithmetic: 3, 7, 11, 15, ? has differences of +4, +4, +4, so the next term is 15 + 4 = 19. Geometric: 3, 6, 12, 24, ? has ratios of x2, x2, x2, so the next term is 24 x 2 = 48. Watch for fractional ratios too. 100, 50, 25, 12.5, ? has a ratio of x0.5 (or divide by 2), so the next term is 6.25. Recursive: 1, 1, 2, 3, 5, 8, ? follows term[n] = term[n-1] + term[n-2], so the next term is 5 + 8 = 13. If neither differences nor ratios are constant, try summing the previous two terms. Second-difference: 2, 6, 12, 20, 30, ? has first differences of 4, 6, 8, 10 and second differences of 2, 2, 2 (constant). The next first difference is 12, so the next term is 30 + 12 = 42. Use this when differences are increasing but not by a constant ratio.

The 30-second solve method

Run the same four tests in the same order on every item. Stop at the first one that fits.

1
Differences (5 seconds)
Compute the differences between consecutive terms. Write them above the sequence. If they're constant, you have an arithmetic sequence. The answer is the last term plus the constant. Done.
2
Ratios (5 seconds)
If differences aren't constant, check the ratios. Divide each term by the previous. If they're constant, you have a geometric sequence. The answer is the last term times the ratio. Done.
3
Recursive (10 seconds)
If neither differences nor ratios are constant, test recursive. Is term[n] = term[n-1] + term[n-2]? Test it on the last two terms. If yes, apply the rule to get the next term.
4
Second differences (10 seconds)
If recursive doesn't fit, compute second differences (the differences of the differences). If they're constant, the first differences are an arithmetic progression. Extrapolate the next first difference and add it to the last term. Done.

When nothing fits If none of the four work in under 30 seconds, mark the item and move on. A small minority of sequence items use exotic patterns like cubes, primes, or alternating rules. They're not worth the time penalty.

Common traps

These four mistakes cost more points than any gap in pattern knowledge. Watch for them on every item.

Trap one: a pattern that fits the last two terms but not the earlier ones. Always verify your candidate pattern against the FIRST two transitions, not just the last. If a pattern only explains terms 4 and 5, it's wrong, even if your computed next term feels right.
Trap two: skipping ahead to the answer. The instinct is to look at the four options and reverse-engineer which one "feels closest." Don't. Compute the pattern explicitly first, then check against the options. The CSC writers craft option distractors specifically to match wrong patterns.
Trap three: alternating sequences. When standard methods all fail, try splitting into odd and even positions. In 2, 5, 4, 7, 6, ? the odd positions (1, 3, 5) are 2, 4, 6 (arithmetic, +2) and the even positions (2, 4) are 5, 7 (arithmetic, +2). The next term is position 6 (even), so 7 + 2 = 9.
Trap four: missing values. Some items show one or two terms blanked. In 3, ?, 12, ?, 48, 96 the ratio is x2 (geometric), so the blanks are 6 and 24.

How to drill

One pattern type per day, twenty items each, then a timed mixed set. Recognize the type before you solve.

1
Day one: arithmetic
Twenty items, arithmetic sequences only. Force yourself to verbalize the difference ("this is arithmetic with d = 5") before computing the answer. Type recognition before solution.
2
Day two: geometric
Twenty items, geometric sequences. Same drill: verbalize the ratio first, including fractional ratios.
3
Day three: recursive
Twenty items, recursive sequences. Most will be Fibonacci-like (sum of previous two), but watch for variants ("each term is double the previous minus 1").
4
Day four: second-difference and alternating
Twenty items. These are the hardest types, so give them their own day.
5
Day five: mixed timed set
Twenty items in eight minutes (24 seconds each, which is exam pace). If you can hit 90%+ at this speed, sequences are fully internalized.

Worked examples

These items are written specifically for this guide. The actual practice bank pulls from a separate pool of original CSE-style items reviewed by passers.

Item 01
Find the next term: 5, 9, 13, 17, 21, ?
- A23
- B24
- C25Correct
- D27
Solution
1. 1
  Compute consecutive differences
  9 − 5 = 4 13 − 9 = 4 17 − 13 = 4 21 − 17 = 4
2. 2
  Identify the pattern
  Constant difference of +4 → arithmetic sequence.
3. 3
  Add the constant to the last term
  21 + 4 = 25
Answer
25
Trap to watch. Arithmetic is the fastest pattern to test, so always start here. Solve time should be under 10 seconds.
Item 02
Find the next term: 2, 6, 18, 54, ?
- A108
- B126
- C162Correct
- D216
Solution
1. 1
  Differences aren't constant
  6 − 2 = 4 18 − 6 = 12 54 − 18 = 36 Not arithmetic.
2. 2
  Test ratios
  6 / 2 = 3 18 / 6 = 3 54 / 18 = 3 Constant ratio of ×3 → geometric sequence.
3. 3
  Multiply the last term by the ratio
  54 × 3 = 162
Answer
162
Trap to watch. Option A (108) is the trap of doubling instead of tripling. Always verify the ratio against ALL transitions before applying it.
Item 03
Find the next term: 1, 1, 2, 3, 5, 8, 13, ?
- A18
- B20
- C21Correct
- D26
Solution
1. 1
  Differences aren't constant
  0, 1, 1, 2, 3, 5. Not arithmetic.
2. 2
  Ratios aren't constant
  1/1=1, 2/1=2, 3/2=1.5, 5/3=1.67... not geometric.
3. 3
  Test recursive (sum of previous two)
  1 + 1 = 2 ✓ 1 + 2 = 3 ✓ 2 + 3 = 5 ✓ 3 + 5 = 8 ✓ 5 + 8 = 13 ✓ Fibonacci sequence.
4. 4
  Apply the rule to get the next term
  8 + 13 = 21
Answer
21
Trap to watch. Option D (26) is the trap of doubling the last term. When standard difference/ratio tests fail, always test recursive (sum or product of previous terms) before giving up.
Item 04
Find the next term: 3, 7, 13, 21, 31, ?
- A41
- B43Correct
- C45
- D47
Solution
1. 1
  Compute first differences
  7 − 3 = 4 13 − 7 = 6 21 − 13 = 8 31 − 21 = 10 First differences: 4, 6, 8, 10. Not constant.
2. 2
  Compute second differences
  6 − 4 = 2 8 − 6 = 2 10 − 8 = 2 Second differences: 2, 2, 2. Constant.
3. 3
  Extrapolate the next first difference
  10 + 2 = 12 (next first difference)
4. 4
  Add to the last term
  31 + 12 = 43
Answer
43
Trap to watch. When first differences are increasing but not by a constant ratio, take second differences. If those are constant, you have a second-difference pattern.
Item 05
Find the next term: 100, 50, 25, 12.5, ?
- A6.25Correct
- B5
- C0
- D10
Solution
1. 1
  Differences aren't constant
  −50, −25, −12.5. Not arithmetic.
2. 2
  Test ratios
  50 / 100 = 0.5 25 / 50 = 0.5 12.5 / 25 = 0.5 Constant ratio of ×0.5 → geometric sequence.
3. 3
  Multiply the last term by 0.5
  12.5 × 0.5 = 6.25
Answer
6.25
Trap to watch. The trap is to compute differences (which are decreasing) and conclude there's no clean pattern. Always check BOTH addition AND multiplication patterns. Fractional ratios are common.
Item 06
Find the next term: 2, 5, 4, 7, 6, 9, ?
- A7
- B8Correct
- C10
- D11
Solution
1. 1
  All four standard tests fail
  Differences: 3, −1, 3, −1, 3. Not constant. Ratios: not constant. Recursive: no clean rule. Second differences: not constant. Try alternating split.
2. 2
  Split into odd-position terms
  Positions 1, 3, 5, 7: 2, 4, 6, ? Arithmetic with +2 → next is 8
3. 3
  Verify with even-position terms
  Positions 2, 4, 6: 5, 7, 9 Arithmetic with +2, a consistent pattern. The 7th term is an odd-position term.
Answer
8
Trap to watch. Alternating sequences look chaotic until you split them by position. When the four standard patterns all fail within 30 seconds, try the split before guessing.
Item 07
Find the next term: 1, 4, 9, 16, 25, ?
- A30
- B32
- C36Correct
- D49
Solution
1. 1
  Recognize the pattern
  1 = 1² 4 = 2² 9 = 3² 16 = 4² 25 = 5² Perfect squares.
2. 2
  Compute the next square
  6² = 36
Answer
36
Trap to watch. Alternative method: second differences are constant (3, 5, 7, 9 → +2 each), so the next first difference is 11, giving 25 + 11 = 36. Both methods work; recognizing perfect squares is faster. Memorize squares up to 15², cubes up to 8³, and Fibonacci numbers.

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Study tactics that actually move the score

01
Always test in order: arithmetic (differences) → geometric (ratios) → recursive → second-difference → alternating. The order matters because earlier types are faster to test and more common.
02
Verify your pattern against the EARLIEST transitions, not just the last. A pattern that fits the last two terms but breaks at the first is the wrong pattern.
03
Watch for fractional ratios. "100, 50, 25, ..." has a ratio of ×0.5, which is easy to miss if you only test integer multipliers.
04
Recognize famous sequences. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100), perfect cubes (1, 8, 27, 64, 125), and Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21) appear directly on the exam.
05
If the four standard tests fail within 30 seconds, try the alternating-sequence split before giving up. A minority of sequence items use this structure.

Frequently asked questions

How many number sequence items appear on the exam?

The CSC does not publish a per-topic count, so nobody can give you an exact number. What is official: the Professional paper has 170 items total and the Subprofessional paper has 165, and number sequence is one of the three listed Numerical Ability topics (basic operations, number sequence, word problems). Treat any specific count you see online as an estimate. The good news is that these items are among the easiest to score well on once you know the four patterns.

Are there sequences that mix multiple patterns?

Rarely. CSC sequences almost always use a single clean pattern. If you find yourself stacking conditions ("the differences double, then minus 3, then square"), you've overcomplicated it. The right pattern is almost always one of the four standard types.

Should I memorize the first 20 perfect squares and cubes?

The first 12-15 squares (up to 15² = 225) and the first 6-8 cubes (up to 8³ = 512), yes. They appear directly on the exam and recognition is much faster than the second-difference method when the sequence is squares or cubes.

What if there are missing terms in the middle, not just at the end?

Same method. Find the pattern using the visible terms (skipping the gap), then back-fill the missing values. In "3, ?, 12, 24, ?, 96" the ratio is x2, so the blanks are 6 and 48.

Can I skip number-sequence items if I'm running short on time?

The method is fast enough (sub-30-second per item) that with practice you should rarely need to skip. If you are tight on time, skip word problems first, since they cost more seconds per item than a clean sequence.

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The four pattern types and how to spot each one

Type

Rule

How to test

Example

Arithmetic

Each term differs from the previous by a constant

Differences are constant

3, 7, 11, 15 (+4 each)

Geometric

Each term is the previous times a constant ratio

Ratios are constant

3, 6, 12, 24 (x2 each)

Recursive (Fibonacci-like)

Each term is built from the previous one or two

Sum of the previous two terms fits

1, 1, 2, 3, 5, 8

Second-difference

The differences themselves form a pattern

Differences of differences are constant

2, 6, 12, 20, 30