Number Series

The most common Prelims series type relies on differences between consecutive terms. Always compute first differences first. If they are constant, the series is arithmetic (e.g. +5, +5, +5). If they grow, check the SECOND differences. Patterns frequently seen in IBPS PO:

• Constant addition/subtraction: 7, 12, 17, 22 (+5)
• Increasing difference (AP of differences): 3, 4, 6, 9, 13 (+1, +2, +3, +4)
• Difference itself doubling: 2, 3, 5, 9, 17 (+1, +2, +4, +8)
• Mixed +/- alternation: 10, 8, 11, 7, 12 (-2, +3, -4, +5)

Key exam tip: write the differences in a row BELOW the series. 80% of wrong-number questions become obvious once differences are stacked. Do not over-think; banks reward the candidate who eliminates the pattern in under 20 seconds and moves on.

Layered-difference method (speed trick):

Layer 1 = a2-a1, a3-a2, a4-a3, ...
Layer 2 = differences of Layer 1

Decision rule:

• Layer 1 constant -> add that constant for next term
• Layer 2 constant -> next Layer-1 diff = last diff + Layer-2 constant; add to last term
• Layer 1 ratios constant (x2, x3) -> multiplicative pattern, not pure difference

Useful prime/square/cube anchors to memorise for instant recognition:
Squares: 1,4,9,16,25,36,49,64,81,100,121,144
Cubes: 1,8,27,64,125,216,343,512
Primes: 2,3,5,7,11,13,17,19,23,29,31,37

When a number sits near n^2 or n^3 +/- a small constant, suspect a square/cube series rather than a difference series.

Series: 4, 6, 10, 18, 33, 66, 130

Step 1 - stack differences: 2, 4, 8, 15, 33, 64.
These are messy, so test the doubling rule instead: each term = previous x2, then -2, -2... Check: 4x2-2=6 (yes), 6x2-2=10 (yes), 10x2-2=18 (yes), 18x2-2=34 — but series shows 33. So 33 is the candidate error; should be 34.

Verify forward: 34x2-2=66 (matches), 66x2-2=130 (matches). Pattern is x2 - 2 throughout.

Answer: 33 is wrong; it should be 34.

Speed note: when you see a number almost double the previous, immediately test 'x2 +/- constant' before drowning in differences. This saves 15-20 seconds per question.

When terms grow fast (roughly doubling, tripling, or more), the engine is usually multiplication, not addition. Three families dominate IBPS PO Mains:

1. Constant ratio (GP): 3, 6, 12, 24 (x2)
2. Increasing-multiplier: 2, 2, 4, 12, 48 (x1, x2, x3, x4)
3. Multiply-then-adjust: 5, 11, 23, 47 (x2 + 1)

The trap: a 'x increasing factor' series produces numbers that look chaotic. The fix is to divide each term by its predecessor and read off the ratio sequence (1, 2, 3, 4 ...). Fractional ratios like x1.5 or x2.5 are common in harder Mains sets, so do not assume integer multipliers. Always sanity-check the LAST given term against your rule before answering.

Core rule for increasing-multiplier series:
a_{n+1} = a_n x k, where k = 1, 2, 3, 4, ... (or 0.5, 1, 1.5, 2 ...)

Fast detection:

• Divide consecutive terms. If quotients form 1,2,3,4 -> increasing multiplier.
• If quotients are 1.5, 2, 2.5, 3 -> half-step multiplier (very common in Mains).
• If a quotient is non-integer but consistent (x2.5), keep it — banks DO use 2.5.

Multiply-and-add detection:
a_{n+1} = a_n x c +/- d. Test c=2 first (most frequent), then c=3.
Find d from the first pair, confirm on the second pair.

Memory aid 'DR-MA': Divide for Ratio, then Multiply-Add. Run divide-test before any addition-test on fast-growing series — it resolves most questions in one pass.

Series: 6, 9, 18, 45, 135, ?

Step 1 - divide consecutive terms: 9/6 = 1.5, 18/9 = 2, 45/18 = 2.5, 135/45 = 3.
The multiplier rises by 0.5 each time: 1.5, 2, 2.5, 3, so next = 3.5.
Step 2 - apply: 135 x 3.5 = 472.5.

Answer: 472.5.

Why this matters: a candidate testing only integer multipliers gets stuck. The half-step ladder (1.5, 2, 2.5, 3, 3.5) is an IBPS PO Mains favourite. The instant you see a x1.5 first ratio, expect a +0.5 ladder and you can predict the answer before fully computing — then just do the final multiplication.

A large share of IBPS PO series are disguised square or cube sequences. The terms look irregular until you map them to n^2 or n^3 with a small adjustment. Common disguises:

• n^2 +/- k: 3, 8, 15, 24, 35 = 2^2-1, 3^2-1, 4^2-1, 5^2-1, 6^2-1
• n^3 +/- k: 0, 7, 26, 63 = 1^3-1, 2^3-1, 3^3-1, 4^3-1
• n^2 + n: 2, 6, 12, 20, 30 = 1x2, 2x3, 3x4, 4x5
• Alternating squares and cubes

The single most valuable exam asset is instant recall of squares to 30 and cubes to 15. When a term is close to a perfect square or cube, subtract the nearest power and look for a clean constant. Do not grind differences first on irregular fast series — try the power map.

Must-memorise tables for speed:
Squares 11-20: 121,144,169,196,225,256,289,324,361,400
Squares 21-30: 441,484,529,576,625,676,729,784,841,900
Cubes 6-12: 216,343,512,729,1000,1331,1728

Key identities used as series rules:

• n^2 - 1 = (n-1)(n+1): 3,8,15,24,35,48,63
• n(n+1): 2,6,12,20,30,42,56
• n(n+2) or n^2+n style products
• Sum of consecutive cubes/squares

Detection trick: if alternate terms jump hugely, suspect alternating series (one sub-series squares, the other cubes or arithmetic). Split odd-position and even-position terms and analyse each separately — this 'alternate split' cracks most Mains power series in seconds.

Series: 0, 3, 8, 15, 24, ?, 48

Step 1 - test the power map. 0 = 1^2-1, 3 = 2^2-1, 8 = 3^2-1, 15 = 4^2-1, 24 = 5^2-1.
Step 2 - the missing term is 6^2-1 = 35, and the last given 48 = 7^2-1 confirms the rule.

Answer: 35.

Cross-check with differences: 3, 5, 7, 9, 11, 13 — consecutive odd numbers, which is exactly the signature of an n^2-based series (since n^2 increments by successive odd numbers). Either route works; the power map is faster once you recognise that the differences are odd numbers in arithmetic progression. Train your eye to flag '3,5,7,9...' differences as a square series immediately.

IBPS PO Mains shifted from 'find next term' to two harder formats:

1. Find the wrong number: a full series is given with one rogue term. Establish the rule from the FIRST 3-4 clean terms, then march forward; the first term that violates the rule is your answer. Never assume the wrong term is in the middle.

2. Two-line (parallel) series: Series I is fully given with a rule; Series II starts at a stated value and follows the SAME rule. You must find a specified term (often the 3rd or 4th) of Series II. Solve Series I to extract the operation pattern, then apply it from Series II's starting number.

Discipline beats cleverness here: lock the rule on the early terms, write each operation explicitly, and apply identically to the second line. Mis-reading which line to extend is the commonest avoidable error.

Two-line series solving framework:

Step 1 (Series I): list operations o1, o2, o3 between successive terms. Common operation chains:
+/- successive squares: +1, +4, +9, +16
x then +: x2+1, x2+2, ...
+/- alternating constants
Step 2: write operations as a portable recipe, e.g. [x2, then +k that itself increases].
Step 3 (Series II): apply o1, o2, o3 from the given start term.

Wrong-term checklist:

• Compute forward from term1 using the locked rule.
• The FIRST mismatch is the wrong term (later terms may be 'built on' the error in well-set papers, but standard sets keep only one rogue value).
• Verify the corrected value restores the chain end-to-end.

Time target: Series I rule in ~25s, Series II answer in ~15s.

Series I: 4, 6, 12, 30, 90, 315
Series II starts at 6. Find the 4th term of Series II.

Step 1 - extract operations from Series I:
6/4 = 1.5, 12/6 = 2, 30/12 = 2.5, 90/30 = 3, 315/90 = 3.5.
Multipliers: x1.5, x2, x2.5, x3, x3.5 (half-step ladder).

Step 2 - apply to Series II from 6:
Term1 = 6
Term2 = 6 x 1.5 = 9
Term3 = 9 x 2 = 18
Term4 = 18 x 2.5 = 45

Answer: the 4th term of Series II is 45.

Lesson: the operation recipe is portable. Extract it cleanly from the fully-given line, then run the SAME multipliers from the new start. Count terms carefully — 'find the 4th term' means three operations, not four.