A number series is a sequence of numbers arranged according to a hidden rule. Your job is either to find the next term, identify the missing term inside the series, or spot the one term that breaks the pattern (the "wrong term" variant).
Think of it like a cipher. The examiner encodes a rule — add, multiply, square, cube, combine operations — and your job is to decode it faster than the other candidates. The terms you see are clues; the pattern is the answer.
Here is a useful way to think about it: imagine each term as a step in a staircase. If you can figure out the size of each step (or how the step-size itself changes), you can predict any future step. Sometimes the staircase has uniform steps (constant difference), sometimes the steps double (geometric progression), sometimes the step-sizes themselves form a second staircase (second-order differences), and sometimes the architect got creative and combined two operations at once.
Why does this matter more in SBI PO than other exams? SBI PO sets series that are one layer deeper than standard. A plain geometric progression rarely appears by itself — it is disguised behind an added constant (aₙ × 2 + 1) or the series alternates between two interleaved sub-sequences. If you only know one pattern type, you will be stuck on precisely those questions.
The good news: every series in the exam is deterministic. There is no ambiguity once you find the rule. That means if you build a mental library of the seven core pattern families and a disciplined first-move habit, you can crack most series in under 30 seconds.
1. Constant Difference (AP)
Each term differs from the previous by the same fixed number.
Example: 2, 5, 8, 11, 14 — difference is always +3.
Spot it: first differences are all equal.
2. Constant Ratio (GP)
Each term is a fixed multiple of the previous.
Example: 5, 20, 80, 320 — ratio is always ×4.
Spot it: divide consecutive terms; the ratio stays constant.
3. Doubling / Halving Differences
The differences between terms form their own GP (usually ratio 2).
Example: 10, 17, 31, 59, 115 — differences are 7, 14, 28, 56 (each doubled).
Spot it: write out differences; check if those differences themselves multiply by a constant.
4. Increasing / Decreasing Differences (Second-Order AP)
Differences form an AP, not a GP.
Example: 3, 15, 35, 63, 99 — differences are 12, 20, 28, 36 (each increases by +8).
Spot it: first differences form an AP with their own constant gap.
5. Two-Operation Rule (×n + c or ×n − c)
Each term is obtained by multiplying the previous by a number and then adding (or subtracting) a constant.
Example: 1, 4, 13, 40, 121 — rule is ×3 + 1.
Example: 11, 23, 47, 95, 191 — rule is ×2 + 1.
Spot it: differences aren't clean, but ratios are close to integers. Test aₙ × 2 + 1 or aₙ × 3 + 1 on the first pair; if it holds, verify on the second pair.
6. Powers of n (Squares, Cubes, Primes)
Terms are n², n³, or pₙ² (squares of primes).
Example: 1, 8, 27, 64, 125 — 1³, 2³, 3³, 4³, 5³.
Example: 4, 9, 25, 49, 121 — 2², 3², 5², 7², 11² (prime squares).
Spot it: check if terms are perfect squares or cubes. If squares don't match natural numbers, check primes.
7. Factorial / Consecutive Multiplier
Each term is the previous term multiplied by the next integer in sequence.
Example: 6, 12, 36, 144, 720 — multiply by 2, 3, 4, 5.
Spot it: ratios between consecutive terms are 2, 3, 4, 5... — consecutive integers.
Step 1 — Write the first differences. Always do this first. It takes 5 seconds and immediately tells you whether you're in family 1, 3, 4, or needs further analysis.
Step 2 — If differences aren't constant, check ratios. Divide consecutive terms. If the ratio is constant → GP. If the ratio is increasing integers → factorial family.
Step 3 — If neither works, check for the two-operation rule.
Take T₂ / T₁ → gives you an approximate multiplier m. Then check: T₁ × m + c = T₂. Solve for c. Verify on T₂, T₃.
Step 4 — Check if terms are recognisable powers.
Mentally run 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 (squares) and 1, 8, 27, 64, 125, 216 (cubes). If a term matches, check the whole series.
Step 5 — For wrong-term questions, find the term that breaks the pattern you've identified, not the term that looks odd at first glance. Sometimes the "odd-looking" term is correct and the "normal-looking" one is wrong.
A harder SBI PO variant interleaves two separate sequences:
2, 3, 6, 9, 18, 27, 54, ...
Here, odd-positioned terms: 2, 6, 18, 54 (multiply by 3) and even-positioned: 3, 9, 27 (multiply by 3). Solve each alternate sub-series independently.
Write differences first (takes 5 seconds). If they're constant → done. If they double → doubling-difference family. If they increase by a fixed amount → second-order AP. If none of those work, compute ratios. This two-move sequence eliminates 5 out of 7 pattern families in under 10 seconds.
Standard approach (trial-and-error each type): ~60 seconds. This structured two-move check: ~10 seconds to narrow to one or two families.
When you suspect ×m + c, test only on the first pair to get c, then verify on the second pair. Never test all pairs upfront.
Formula: c = T₂ − (T₁ × m) where m is the nearest integer to T₂ / T₁.
Example: 11, 23 → 23 / 11 ≈ 2.09 → try m = 2, c = 23 − 22 = 1. Verify: 23 × 2 + 1 = 47. Confirmed in 2 steps instead of testing every pair. Saves 3-4 steps.
Keep this list active in memory: 4, 9, 25, 49, 121, 169, 289, 361. If you see any two of these in a series, immediately check whether all terms are prime squares — you don't need to check ratios or differences at all.
Standard method (compute differences, get confused): ~45 seconds. Recognition from checklist: ~8 seconds.
For the factorial/consecutive-multiplier family, compute T₂/T₁, T₃/T₂, T₄/T₃. If you get 2, 3, 4 (or 3, 4, 5, etc.), next ratio is the next integer. Multiply the last known term by that integer.
Example: 6, 12, 36, 144 → ratios: 2, 3, 4 → next ratio 5 → 144 × 5 = 720. This is a 2-second final step once you spot the ladder. Standard method: manually continuing the multiplication table → ~30 seconds.
Once you identify the pattern and compute the answer, check the units digit of your result against the options before doing full arithmetic. For doubling-difference series, the units digit cycles predictably. For a series ending in 115 + 112, units digit of answer = 5 + 2 = 7 → only 227 has units digit 7 among the options. Choose it without writing 115 + 112 fully.
This eliminates computation errors and saves ~10 seconds per question.
In the exam hall, follow this decision tree rigidly. Don't browse randomly.
Move 1 (5 sec): Write first differences. Are they equal? → AP, done.
Move 2 (5 sec): Do differences double/halve? → Doubling-difference series, done.
Move 3 (5 sec): Do differences themselves form an AP? → Second-order AP, done.
Move 4 (5 sec): Compute consecutive ratios. Constant? → GP, done. Consecutive integers (2,3,4…)? → Factorial multiplier, done.
Move 5 (8 sec): Try ×m + c. Take T₂ / T₁, round to nearest integer for m, solve for c, verify on next pair.
Move 6 (5 sec): Are terms recognisable squares, cubes, or prime squares? Scan checklist.
Move 7 (5 sec): Check for interleaved two-sequence. Separate odd and even indexed terms, apply moves 1-6 to each.
If none of these yield a clear rule within 45 seconds, use elimination — compute what each option implies as the rule and check if it holds for all given terms. Budget no more than 90 seconds total per series question. If it's still unclear, flag and move on.
Why this question: Tests doubling-difference pattern — the most frequently disguised type in SBI PO. Students often try to find a ratio and get confused.
Solving path: Write differences: 17−10=7, 31−17=14, 59−31=28, 115−59=56. These differences are 7, 14, 28, 56 — each is double the previous. Next difference: 56×2=112. Answer: 115+112=227. Units digit check: 5+2=7 → matches only option A. Time: ~20 seconds.
Why this question: Tests the consecutive-multiplier (factorial) family, which students often misread as a simple GP because the first ratio 12/6=2 looks geometric.
Solving path: Ratios: 12/6=2, 36/12=3, 144/36=4. The ratio ladder is 2, 3, 4 — next ratio must be 5. Answer: 144×5=720. If you stopped at "ratio≈2" and tested GP with constant ratio, you'd get 144×2=288 — not in the options, which is your signal to recheck. Time: ~18 seconds.
Why this question: Tests the ×3 + 1 two-operation rule — the core disguise technique SBI PO uses to move beyond simple AP/GP.
Solving path: Differences: 3, 9, 27, 81 — themselves a GP with ratio 3. Alternatively, check two-operation: 1×3+1=4, 4×3+1=13, 13×3+1=40, 40×3+1=121, 121×3+1=364. Both approaches converge. Units digit: 121×3=363, 363+1=364, units digit 4 → option A only. Time: ~25 seconds.
Why this question: Tests prime-squares recognition. Candidates who only know natural-number squares will call the series wrong because 4, 9, 25 don't fit n² for consecutive n.
Solving path: Check if terms are perfect squares: 4=2², 9=3², 25=5², 49=7², 121=11². Notice the bases: 2, 3, 5, 7, 11 — consecutive primes. Next prime: 13. Answer: 13²=169. Prime squares checklist activated in move 6. Time: ~15 seconds once you recognise the pattern.
Why this question: Tests ×2 + 1 — appears in SBI PO to trap candidates who know ×3 + 1 but haven't practised other multipliers.
Solving path: Ratios: 23/11≈2.09. Try m=2: c = 23 − 22 = 1. Verify: 23×2+1=47 ✓, 47×2+1=95 ✓, 95×2+1=191 ✓. Final term: 191×2+1=383. Units digit: 191×2=382, 382+1=383, units digit 3 → option A. Time: ~22 seconds.
Stopping at first differences without checking second-order. If the first differences aren't constant, always check whether they form an AP before concluding the series is non-arithmetic. Many SBI PO series use second-order differences, and candidates misclassify them as two-operation series, wasting 40 extra seconds.
Mistaking a consecutive-multiplier series for GP. The first ratio in a factorial series is 2, which looks like a GP with r=2. Always compute at least two ratios before declaring GP. If ratios are 2, 3, 4 — it is never a GP.
In prime-square series, checking only consecutive natural-number squares. Candidates write 1, 4, 9, 16, 25... and find a mismatch, then abandon the squares angle entirely. Always check prime bases as a second sub-step.
Verifying the two-operation rule on only one pair. You solve c from T₁, T₂, plug into T₂, T₃, it works — and you assume you're done. Always verify on at least two consecutive pairs. A coincidence match on one pair is possible.
In wrong-term questions, changing the term that looks visually out of place. Examiners deliberately put the wrong term in a position where it looks almost right. Identify the rule from the maximum number of consistent terms, then find which single term violates it — don't trust visual intuition here.
Not using units digit elimination on the final step. After identifying the rule, candidates compute the full answer and then read options. Train yourself to compute units digit of your answer first, then immediately scan options for it. This catches arithmetic errors and eliminates 2-3 options instantly, which is especially useful when you are unsure of your full calculation.