How to Solve Number Series Questions: A Step-by-Step Method

“Number sequence” is one of the three items in the official Numerical Ability scope of the Civil Service Exam. Series questions look like puzzles, but nearly all of them are built from a small set of patterns. If you test those patterns in a fixed order, you will rarely be stuck. This guide shows the method with worked examples.

Step 1: Look at the overall direction

Before calculating anything, ask:

• Are the numbers increasing or decreasing?
• Is the change slow (suggesting addition or subtraction) or fast (suggesting multiplication, squares or powers)?
• Do the numbers go up and down (suggesting two alternating series)?

This takes two seconds and tells you which patterns to try first.

Step 2: Write the differences

Under the series, write the difference between each pair of neighbouring terms. This single step solves most items.

Constant difference

7, 12, 17, 22, ___ Differences: +5, +5, +5 → next is 27.

Differences that grow steadily

100, 96, 88, 76, 60, ___ Differences: −4, −8, −12, −16 → next difference −20 → 40.

2, 6, 12, 20, 30, ___ Differences: +4, +6, +8, +10 → next +12 → 42.

If the differences themselves form a pattern, you have found it.

Step 3: If differences fail, try ratios

When numbers grow or shrink fast, divide instead of subtract.

3, 6, 12, 24, ___ → each term × 2 → 48 81, 27, 9, 3, ___ → each term ÷ 3 → 1

Sometimes the rule mixes both:

5, 11, 23, 47, ___ Each term is doubled and 1 is added: 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47 → 95.

A good clue: if each term is roughly double the previous one, test “× 2 plus or minus a small number”.

Step 4: Check for squares and cubes

Learn the squares from 1² to 20² and the cubes from 1³ to 10³. They appear often:

1, 4, 9, 16, 25, ___ → squares → 6² = 36 1, 8, 27, 64, ___ → cubes → 5³ = 125 2, 5, 10, 17, 26, ___ → squares plus 1 (1+1, 4+1, 9+1 …) → 36 + 1 = 37

Step 5: Look for “add the previous two”

In Fibonacci-type series, each term is the sum of the two before it.

3, 4, 7, 11, 18, ___ → 11 + 18 = 29

Step 6: Split alternating series

If the numbers go up and down, take every other term as a separate series.

2, 20, 4, 17, 6, 14, ___ Odd positions: 2, 4, 6 (+2). Even positions: 20, 17, 14 (−3). The missing term is in an odd position → 8.

Step 7: Know the special sequences

A few sequences are worth recognising on sight:

• Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23 …
• Triangular numbers: 1, 3, 6, 10, 15, 21 … (add 2, 3, 4, 5 …)
• Products of consecutive numbers: 2, 6, 12, 20, 30 … (1×2, 2×3, 3×4 …)

Letter series use the same method

Convert letters to their positions in the alphabet (A = 1, B = 2 … Z = 26) and apply the same steps.

A, C, F, J, O, ___ → 1, 3, 6, 10, 15 → differences +2, +3, +4, +5 → next +6 → 21 → U.

Knowing a few letter positions by heart helps: E = 5, J = 10, O = 15, T = 20, Y = 25.

A checklist to keep in your head

1. Direction and speed of change
2. Differences (and differences of differences)
3. Ratios
4. Squares and cubes, plus or minus a constant
5. Sum of the previous two terms
6. Alternating series
7. Special sequences (primes, triangular numbers)

Work through it in order. If nothing fits within about a minute, mark the item, move on, and return later with fresh eyes.

Check your answer

Once you find a rule, test it on every term, not just the last few. A rule that fits only part of the series is wrong.

Practise

Number series improve quickly with practice, because you start to recognise patterns at a glance. Work through the series items in our basic operations and number sequence set and the pattern questions in word analogy and logical reasoning. Each explanation names the rule, so you can add it to your own checklist.