A number series is a sequence of numbers arranged according to a hidden rule. Your job is to find that rule and use it to identify the missing term or the next term.
Think of it like a lock combination. Each dial follows a pattern — once you crack one dial's logic, the rest open up. In number series, every term is the "dial" and the pattern is the combination you need to find.
Here's the key insight that most students miss: the pattern is almost always one of five types, and your first 10 seconds should be spent categorising the series, not calculating.
The five types you will see in UP Police Constable:
Arithmetic (AP): Each term increases or decreases by a fixed number. Example: 3, 7, 11, 15 — difference is always +4.
Geometric (GP): Each term is multiplied or divided by a fixed number. Example: 2, 6, 18, 54 — ratio is always ×3.
Squares / Cubes: Terms are perfect squares (n²) or cubes (n³), sometimes with a twist like n²(n+1).
Difference series: The differences between consecutive terms themselves form a pattern (AP or GP). This is the most common "harder" type.
Mixed / Compound: Two or more rules running together, or alternating rules.
The analogy that sticks: arithmetic series are like uniform steps on a staircase. Geometric series are like a ball bouncing — each bounce is a fixed fraction of the last. Difference series are like acceleration — the speed changes, but the change in speed is constant.
Once you know which staircase you are on, finding the missing step is just arithmetic.
Always start by subtracting consecutive terms. Write the differences below the series.
Series: 42, 45.5, 49, ?, 56, 59.5, 63
Diff: +3.5, +3.5, +3.5, +3.5, +3.5, +3.5
If the differences are constant, you have an AP. Done in 5 seconds.
Divide each term by the previous one.
Series: 0.12, 0.48, 1.92, 7.68, 30.72, ?
Ratios: ×4, ×4, ×4, ×4, ×4
Constant ratio = GP. Missing term = 30.72 × 4 = 122.88.
Calculate differences of the differences.
Series: 326, 452, 678, 1004, ?, 1956, 2582
Diff 1: 126, 226, 326, 426, 526, 626
Diff 2: 100, 100, 100, 100, 100
The second differences are constant at +100. So the first differences are an AP with +100 each step. The missing first difference is 326 + 100 = 426, making the missing term 1004 + 426 = 1430.
Look at the numbers and ask: are these close to perfect squares or cubes?
144 = 12², 169 = 13², 196 = 14², 225 = 15², ? = 16² = 256
When you see numbers like 144, 169, 196 — your brain should fire "squares" immediately, not "let me find the difference."
If none of the above works, try n²(n+1), n(n+1), n³ + n, or similar. These are rare in UP Police but do appear.
Series: 2, 12, 36, 80, ?
Pattern: 1²×2=2, 2²×3=12, 3²×4=36, 4²×5=80, 5²×6=150
Decimal series follow the exact same rules — don't let the decimals intimidate you. Treat 0.8, 1.6, 2.4 like 8, 16, 24 — just with a decimal point. The difference is 0.8 each time, same logic.
For a GP with decimals like 0.12, 0.48, 1.92 — the ratio ×4 works the same way. 30.72 × 4 = 122.88, and you can verify: 122.88 × 4 = 491.52. Always verify with the term after the blank when possible.
This deserves extra attention because it catches most students off guard. When first differences form an AP themselves:
This process is mechanical. You do not need to derive a formula. Just keep building the pattern downward until you hit a constant row, then rebuild upward.
When consecutive differences are equal, the series is AP. Don't calculate anything else. Scan the series: if Term2 − Term1 = Term3 − Term2, add that difference to the term before the blank. Example: 42, 45.5, 49, ___, 56 → differences are all 3.5 → answer is 49 + 3.5 = 52.5. Standard method (trial options): ~40s. This method: ~8s.
Memorise squares from 12² to 20² cold: 144, 169, 196, 225, 256, 289, 324, 361, 400. When you see any of these in the series, immediately write the root below each number (12, 13, 14...) and the pattern becomes trivially obvious. Standard method (find differences): 3 steps. This method: 1 recognition + 1 step. Saves 2 steps per question.
If subtracting gives non-constant differences, try dividing term2 ÷ term1. Do it for two pairs. If ratio matches, it's GP. Stop there, multiply the last known term by the ratio. For 0.12 → 0.48 → 1.92: 0.48 ÷ 0.12 = 4, 1.92 ÷ 0.48 = 4. Confirmed. 30.72 × 4 = 122.88 in one multiplication. If you tried subtraction first: non-constant, confused, then GP check — adds ~20s. Going GP-check-first when numbers grow rapidly: saves ~15s.
When first differences are not constant, write them out and subtract again. If the second-level differences are constant, rebuild the answer going upward. For series 326, 452, 678, 1004, ?, 1956: differences are 126, 226, 326, 426 — these increase by 100 each time. Next difference = 426 + 100 = 526... wait, 1004 + 426 = 1430. Verify: 1430 + 526 = 1956. Correct. This 2-level ladder check resolves what looks like a "hard" series in ~25s versus 60s+ of random formula-guessing.
In decimal series questions, check the decimal place of the correct answer before calculating. If the series is in tenths (0.8, 1.6, 2.4...) the answer must also be in tenths. If three options have different decimal places from the series pattern, eliminate them instantly. In the 0.8, 1.6, 2.4, 3.2, ___ question: options 3.8, 5.4, 4.6 are tenths but wrong values — only 4.0 continues the exact +0.8 pattern. Recognising the pattern first means you don't even need to check 3.8, 5.4, or 4.6 arithmetically. Saves ~10s per such question.
In the exam hall, use this exact decision tree. Do not skip steps.
Step 1 (5s): Scan the series. Are the numbers growing? Shrinking? Alternating? This tells you the sign of the pattern.
Step 2 (5s): Calculate two consecutive differences. Are they equal? If yes → AP. Find the missing term. Move on.
Step 3 (5s): If differences are unequal, calculate two consecutive ratios. Are they equal? If yes → GP. Find the missing term. Move on.
Step 4 (10s): Calculate all first differences. Do these differences themselves form an AP? If yes → difference ladder. Add the next step in the difference AP, then add to the last term.
Step 5 (5s): If none of above, check if terms are perfect squares or cubes in sequence.
Step 6 (10s): Try formula patterns: n²(n+1), n(n+1)/2, n³ + n. These are rare but appear.
Fallback (20s): If completely stuck, plug the answer options back into the series and verify. Start with the middle option. If it makes the surrounding pattern consistent, that is your answer.
Total target time per number series question: under 45 seconds.
Why this question: This is the simplest AP decimal series that appears in UP Police. Many students second-guess themselves because of the decimal — this question trains you not to.
Solving path: Calculate difference: 1.6 − 0.8 = 0.8. Verify: 2.4 − 1.6 = 0.8. Confirmed AP with +0.8. The term before the blank is 3.2. Answer: 3.2 + 0.8 = 4.0. Check: 4.0 + 0.8 = 4.8 (matches next given term). Option D is correct.
Why this question: Tests GP recognition with decimals. Students often try subtraction first and waste 20 seconds. Train yourself to switch to division when the series is growing rapidly.
Solving path: Ratios: 0.48 ÷ 0.12 = 4, 1.92 ÷ 0.48 = 4. GP with ratio 4. Missing term = 30.72 × 4 = 122.88. Verify: 122.88 × 4 = 491.52. Option C is correct.
Why this question: Another AP with decimals (actually 3.5 step), tests whether you can handle non-integer common differences calmly.
Solving path: Difference: 45.5 − 42 = 3.5. Verify: 49 − 45.5 = 3.5. Missing term = 49 + 3.5 = 52.5. Verify: 56 − 52.5 = 3.5. Option B is correct.
Why this question: Classic second-difference series. This is where most students get stuck. This question is exactly the type that separates 60-percentile from 80-percentile scorers.
Solving path: First differences: 452−326=126, 678−452=226, 1004−678=326. Second differences: 226−126=100, 326−226=100. Second differences are constant at 100. So next first difference = 326 + 100 = 426. Missing term = 1004 + 426 = 1430. Verify: 1956 − 1430 = 526 = 426 + 100. Option A is correct.
Why this question: Perfect square series — the most important pattern to recognise on sight. Appears regularly and is solvable in under 10 seconds if you know your squares.
Solving path: Recognise: 144 = 12², 169 = 13², 196 = 14², 225 = 15². Pattern is consecutive perfect squares. Next term = 16² = 256. Option D is correct.
Why this question: Compound formula n²(n+1). This tests whether you go beyond simple AP/GP when the series doesn't fit either. The formula pattern is elegant and repeats across exams.
Solving path: Differences: 12−2=10, 36−12=24, 80−36=44. Second differences: 24−10=14, 44−24=20. Not constant. Try formula: n=1 → 1²×2=2, n=2 → 4×3=12, n=3 → 9×4=36, n=4 → 16×5=80, n=5 → 25×6=150. Pattern confirmed. Answer: 150. Option A is correct.
Subtracting when you should be dividing. When terms grow very fast (0.12 → 0.48 → 1.92), the differences are also growing fast and look irregular. Always try the ratio check when the differences are not constant. Don't spend more than 5 seconds on subtraction before switching.
Ignoring the term after the blank. Every problem gives you terms on both sides of the blank. Use the term after the blank as a verification check. If your answer doesn't produce the next term correctly when you apply the pattern forward, your pattern identification is wrong — go back.
Panicking at decimal series. A series in decimals follows the exact same five types as a whole-number series. Multiply everything by 10 or 100 mentally to clear the decimal if it helps — 0.8, 1.6, 2.4 becomes 8, 16, 24, which is obviously +8 each time. Then re-apply the decimal.
Stopping at first differences when they're not constant. This is the biggest mistake in harder questions. When first differences are 126, 226, 326 — those are not "random," they themselves differ by 100. Always compute second differences before giving up on a series.
Assuming every hard-looking series is GP. Not everything growing fast is geometric. Check ratio for exactly two pairs before committing. If the ratio differs between pairs 1-2 and 2-3, it is not a simple GP — move to difference ladder or formula.
Not verifying with given terms. If the series has 7 or 8 terms and you find the missing one, use the two terms surrounding the blank to verify both sides. One-sided verification misses half the check.