Series (Number & Letter)
🔒 Log in to trackHow every series is cracked
🔒 Log in to trackA series question shows 5-6 terms and asks for the next (or a missing middle) term. Every CGL series is built by one small rule — your job is to name it before the clock names it for you.
The universal method: the difference ladder. Subtract neighbours. If the differences are constant, you have an AP. If not, subtract again (second differences). Repeat until a row is constant, then climb back up. Most CGL series die at the second row. If differences look meaningless, switch families: multiply (GP), alternate, or squares/cubes.
Reading the ladder tells you the family instantly:
| What you see | Family to try |
|---|---|
| Differences constant | AP (add k) |
| Second differences constant | based on n² |
| Each term ≈ previous × k | GP / ×k ± r |
| Ratios of differences constant | ×k on differences |
| Two clean mini-rows (odd/even positions) | interleaved series |
| Differences are 1, 8, 27... or 1, 3, 5... | cubes / odd numbers added |
Detailed notes
What is a series?
A series is a list of numbers or letters written in a fixed order. Every term is made from the one before it by the same small rule. Think of a savings box: you put ₹100 in January, ₹150 in February and ₹200 in March. Anyone can tell that April will be ₹250, because the rule "add ₹50 every month" is clear. A series question hides such a rule and asks you to find it.
SSC, Railway and Banking papers ask series in three formats:
- Next term: '4, 9, 16, 25, ?' — find what comes after the last term.
- Missing term: '4, 9, ?, 25, 36' — a term in the middle is replaced by '?'.
- Wrong term: the series is complete but one term is incorrect (see the Wrong number lesson).
Step 1: Look at the size of the numbers
Before any calculation, see how fast the terms grow.
- Slow, steady growth (5, 9, 13, 17) → the rule adds something.
- Terms roughly doubling or tripling (3, 7, 15, 31) → the rule multiplies (maybe with a small add or subtract).
- Terms going up and down (8, 20, 10, 18, 12) → two series are mixed (alternate terms).
- Terms close to famous numbers (3, 8, 15, 24 are just below 4, 9, 16, 25) → squares or cubes.
Step 2: The difference ladder
Write the gaps (differences) between neighbours under the series. This is the first difference row.
Example: 2, 5, 10, 17, 26 → differences 3, 5, 7, 9. They rise by 2, so the next difference is 11 and the next term is 26 + 11 = 37.
If the first row is not clear, take differences again (second difference row). Most exam series become clear within two rows.
| What the ladder shows | Family |
|---|---|
| First differences equal | Add the same number (AP) |
| First differences rise evenly | Growing difference |
| Differences double | Powers of 2 are being added |
| Differences are 1, 4, 9, 16 | Squares are being added |
| Each term ≈ previous × k | Multiply family (×k ± r) |
Step 3: When the ladder fails
Try, in this order: (1) ratio — divide a term by the one before it; (2) alternate terms — read positions 1, 3, 5 and 2, 4, 6 separately; (3) squares and cubes near the terms; (4) mixed operations such as ×1+1, ×2+2, ×3+3.
Missing term in the middle
Find the rule from the terms you can see on both sides of the gap. Then check your answer twice: it must come correctly from the term before it, and the term after it must come correctly from your answer.
Example: 3, 7, ?, 31, 63 → each term is ×2 + 1, so ? = 7 × 2 + 1 = 15. Check: 15 × 2 + 1 = 31 ✓.
Fractions, decimals and halving
Do not panic at decimals. 64, 32, 16, 8, 4, ? is just ÷2 → 2. 0.5, 1.5, 4.5, 13.5 is ×3 → 40.5. For fractions, read the top numbers (numerators) and the bottom numbers (denominators) as two separate series: 1/3, 3/5, 5/7 → tops 1, 3, 5 and bottoms 3, 5, 7 both rise by 2, so the next is 7/9.
Two missing terms
Some questions blank two terms and give pairs as options. Find the rule from the visible terms, fill the first blank, then continue to the second blank. Check the term after each blank.
Quick revision
- Look at size first: slow growth = add, fast growth = multiply, up-down = two series.
- Difference ladder: first row, then second row.
- If it fails: ratio → alternate terms → squares/cubes → mixed operations.
- Missing term: check with the term before AND the term after.
- Fractions: numerators and denominators are two separate series.
- Your rule must fit every given term, not just the last two.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Find the next term — choose the family with the difference laddervery common3 practice Q
The question gives 5-7 terms ending in '?'. Nothing is hidden in the middle.
- Look at growth: slow = add, fast = multiply, up-down = two series.
- Write the first difference row; if needed, the second row.
- Continue the row that is regular and climb back up to the next term.
- Check the rule on every given term.
Why it works: every exam series is built by one small repeated rule, and differences expose adding-type rules at once.
Example: 6, 11, 21, 36, 56, ?
Differences: 5, 10, 15, 20 → next difference 25. 56 + 25 = 81.
Type 2: Missing term in the middle of the seriesvery common3 practice Q
A '?' sits between given terms (e.g. 7, 15, ?, 63, 127).
- Find the rule from the terms on both sides of the gap.
- Fill the gap from the term before it.
- Check that the term after the gap follows from your answer.
Why it works: a middle term must fit twice — with its left neighbour and its right neighbour — so the double check removes wrong guesses.
Example: 7, 15, ?, 63, 127
15 = 7×2 + 1 and 127 = 63×2 + 1, so the rule is ×2 + 1. ? = 15×2 + 1 = 31. Check: 31×2 + 1 = 63 ✓.
Type 3: Series with fractions, decimals or halvingcommon3 practice Q
Terms are decimals (0.5, 1.5 ...), fractions, or keep getting halved (240, 120, 60 ...).
- Treat decimals exactly like whole numbers: try ×k or ÷k first.
- For fractions, write the numerators as one series and the denominators as another.
- Build the answer in the same form as the terms.
Why it works: decimals and fractions only change how the numbers look, not the rule behind them.
Example: 81, 27, 9, 3, 1, ?
Each term is ÷3. 1 ÷ 3 = .
Type 4: Two missing terms — choose the correct paircommon3 practice Q
Two '?' marks in one series; each option is a pair of numbers.
- Find the rule from the visible terms.
- Fill the first blank, then keep going to the second blank.
- Check the terms right after each blank.
- Eliminate options where only one of the two numbers is right.
Why it works: both numbers come from the same rule, so one wrong number kills the option.
Example: 5, ?, 15, 20, ?, 30
The rule is +5. First blank = 5 + 5 = 10; second blank = 20 + 5 = 25. Answer 10, 25.
Shortcut tricks
⚡ Difference ladder in 10 seconds
Write the terms with gaps and subtract under them, twice if needed. A constant row anywhere fixes the whole rule; extend the row one step and climb back up to get the answer.
Example: Q. 9, 16, 25, 36, 49, ?
Sol. Differences: 7, 9, 11, 13 → next 15. 49 + 15 = 64. (One glance confirms these are squares 3² to 8² — both readings give 64.)
Ladder first; family names come free afterwards.
Where students lose marks
Extending only the difference row and forgetting to add back to the last term.
Trying exotic rules before finishing the ladder — 80% of CGL series are ladder-friendly.
Assuming the missing term is always last; CGL often hides '?' in the middle, where the ladder must run both ways.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 7 min · wrong answers go to your mistake notebook automatically.