Number Series & Quadratic Equations: Spot the Pattern, Compare the Roots

A from-scratch guide to two of the fastest-scoring topics in IBPS and SBI Prelims — the fixed order for testing number-series patterns (difference, product, square, cube, mixed) and the sign shortcut for comparing the roots of two quadratics (x vs y), with fully worked examples and the common "greater-than-or-equal vs strictly-greater" and "no relation" traps.

By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026

Part of Banking (IBPS / SBI) prep

🧠 You already predict the next station

Think of your daily local train. After a few days you know the order of stations by heart, so the moment the train leaves Dadar you already know Bandra is next, without reading any board. A Number Series works exactly the same way. The examiner gives you a few 'stations' (numbers) and hides one, and your job is to feel the rhythm of how each number leads to the next, then name the missing one. Once you learn the common rhythms, your brain predicts the answer almost automatically. Padho, dost, let's learn those rhythms.

In IBPS and SBI Prelims (both PO and Clerk) the Quant section gives you about 35 questions in just 20 minutes, roughly 34 seconds per question. Number Series and Quadratic Equations are gold here: each set is usually 5 questions, and once you spot the trick a question takes 20 to 30 seconds. But respect the 0.25 negative marking, because one wild guess quietly eats away the marks of a correct answer. So the goal is simple: learn the patterns, solve fast, and skip cleanly when you are not sure.

Part 1: Number Series

A Number Series question shows a row of numbers following one hidden rule, with either a missing term (marked ?) or one wrong term you must catch. There is no single formula here, so you become a pattern detective. The good news is that almost all exam series use just a handful of rhythms, and you test them in a fixed order so you never waste time.

How to find the pattern (test in this order)

  1. 1Write the differences between consecutive terms. Is the gap constant, or is it growing by a fixed amount each time?
  2. 2If differences don't settle it, check the ratio. Is each term roughly double, triple, or half of the previous one? That points to a product series.
  3. 3If the numbers jump fast, test squares and cubes by comparing each term to the nearest perfect square (1, 4, 9, 16...) or cube (1, 8, 27, 64...).
  4. 4Still stuck? Look for a mixed rule like x2+1 or 'multiply by n, add n', or an alternating pattern where odd and even positions follow different rules.
  5. 5Lock a rule only after it fits at least 3 gaps, then apply it to find the missing or wrong term.
PatternHow to spot itExample
DifferenceA constant or steadily growing gap4, 7, 10, 13 (+3 each)
Product / RatioEach term multiplied by a number3, 6, 12, 24 (x2 each)
SquaresTerms sit near 1, 4, 9, 16, 252, 5, 10, 17 (n squared + 1)
CubesTerms sit near 1, 8, 27, 640, 7, 26, 63 (n cubed - 1)
Mixed (xn + n)Grows fast with a changing multiplier5, 6, 14, 45 (x1+1, x2+2...)

📝 Find the missing number: 5, 6, 14, 45, 184, ?

The jump from 45 to 184 is huge, so plain addition won't fit. Suspect multiplication.

Try 'multiply by 1, add 1': 5 x 1 + 1 = 6. It matches.

Then 'multiply by 2, add 2': 6 x 2 + 2 = 14. It matches.

Then x 3 + 3: 14 x 3 + 3 = 45. It matches.

Then x 4 + 4: 45 x 4 + 4 = 184. It matches.

The rule is 'multiply by n, add n', so the next step is x 5 + 5: 184 x 5 + 5 = 925.

Missing number = 925.

💡 Memorise squares up to 30 and cubes up to 15 before exam day. When a series suddenly leaps, like 6 to 45 to 184, stop checking differences and immediately suspect multiplication, squares, or cubes. That one instinct saves you 30 seconds per question.

Part 2: Quadratic Equations (x vs y)

In a Quadratic Equations question you get two equations, one in x and one in y. You solve each to get two roots, then compare the x-values with the y-values and pick the correct relationship. The five options are almost always: x > y, x < y, x is greater than or equal to y, x is less than or equal to y, and 'x = y or no relation can be established'. The whole game is to factorise quickly and then compare the roots without slipping on signs.

For x squared + bx + c = 0 (leading coefficient 1): sum of the two roots = -b, and product of the two roots = c. Shortcut: split the constant c into two numbers whose product is c and whose sum is -b. Those two numbers, each with its sign flipped, are your roots.
Middle term (b)Constant (c)The two roots
+ (positive)+ (positive)Both negative
- (negative)+ (positive)Both positive
+ (positive)- (negative)Opposite signs (one +, one -)
- (negative)- (negative)Opposite signs (one +, one -)

📝 Compare the roots step by step

Given: I) x squared - 5x + 6 = 0 II) y squared - 12y + 35 = 0

Solve I: find two numbers with product 6 and sum 5, which are 2 and 3.

So (x - 2)(x - 3) = 0, giving x = 2 or x = 3.

Solve II: two numbers with product 35 and sum 12, which are 5 and 7.

So (y - 5)(y - 7) = 0, giving y = 5 or y = 7.

Put them on a number line: x-values {2, 3}, y-values {5, 7}.

Largest x = 3, smallest y = 5. Since 3 < 5, every x is smaller than every y.

Answer: x < y.

⚠️ Biggest trap: if the x-values and y-values overlap, say x = {2, 6} and y = {4, 5}, then one x is smaller and one x is bigger than the y-values, so no fixed relation holds. The answer is 'cannot be established', NOT x = y. Use x = y only when both equations give the exact same pair of roots. And when the two groups just touch at one point (largest x equals smallest y, or largest y equals smallest x), use 'greater than or equal to' / 'less than or equal to', never the strict > or <.

Exam-day quick recall

  • ✓Number Series order: difference, then ratio, then squares/cubes, then mixed rule.
  • ✓A sudden big jump means think multiplication or powers, not addition.
  • ✓Quadratics: sum of roots = -b, product = c; split the constant to factorise in seconds.
  • ✓Sign rule: a + constant means the two roots share a sign (the sign opposite to the middle term); a - constant means the roots have opposite signs.
  • ✓Compare largest-x with smallest-y and smallest-x with largest-y to fix the relation.
  • ✓Groups touch at exactly one value: use 'greater/less than or equal to'. Groups overlap: 'cannot be established'.
  • ✓With 0.25 negative marking, skip anything you can't crack in about 40 seconds instead of guessing.

⚡ Quick check

Given I) x squared - 9x + 20 = 0 and II) y squared - 7y + 12 = 0, what is the relationship between x and y?

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