Episode 8 — Aptitude and Reasoning / 8.24 — Series

8.24 Quick Revision -- Series

Series Types at a Glance

Type	Pattern	Example
Arithmetic	Constant difference	3, 7, 11, 15 (d=4)
Geometric	Constant ratio	2, 6, 18, 54 (r=3)
Difference	Differences form a pattern	1, 3, 7, 13 (D1: 2,4,6)
Multiplier	Changing multiplier	1, 2, 6, 24 (x2, x3, x4)
Square-based	n^2, n^2+k, n^2-k	2, 5, 10, 17 (n^2+1)
Cube-based	n^3, n^3+k, n^3-k	0, 7, 26, 63 (n^3-1)
Fibonacci	Sum of 2 previous	1, 1, 2, 3, 5, 8
Prime-based	Uses prime numbers	2, 3, 5, 7, 11, 13
Alternate	Two interleaved series	3, 4, 9, 16, 27, 64
Mixed ops	Alternating operations	5, 6, 14, 15, 45, 46
Power	n^n or similar	1, 4, 27, 256 (n^n)

The Difference Method (Universal Approach)

  Series:  a1  a2  a3  a4  a5
  D1:        d1  d2  d3  d4      <- Subtract consecutive terms
  D2:          e1  e2  e3        <- Subtract D1 terms
  D3:            f1  f2          <- If needed
  
  When you reach a constant level, work backwards to find the answer.

Key Numbers to Memorize

Squares (1-20): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

Cubes (1-10): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Primes (first 15): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

Decision Tree (Quick Pattern Recognition)

  1. Slow growth?     -> Check DIFFERENCES (arithmetic)
  2. Fast growth?     -> Check RATIOS (geometric)
  3. Near squares?    -> Try n^2 +/- k
  4. Near cubes?      -> Try n^3 +/- k
  5. Chaotic?         -> SPLIT odd/even positions
  6. Sum of prev 2?   -> Fibonacci-like
  7. Primes?          -> Check prime series
  8. None above?      -> Mixed operations or x2+1, x3-1 type

Common Patterns

  x2 + 1:   1, 3, 7, 15, 31, 63        (2^n - 1)
  x2 - 1:   1, 1, 1, 1, 1              (converges to 0)
  x3 - 1:   2, 5, 14, 41, 122
  n(n+1):   2, 6, 12, 20, 30, 42       (consecutive products)
  n^2 + 1:  2, 5, 10, 17, 26, 37
  n^3 - 1:  0, 7, 26, 63, 124, 215
  Factorial: 1, 1, 2, 6, 24, 120, 720

Wrong Number Strategy

Find the pattern the series SHOULD follow
Compute correct terms
The one that doesn't match is wrong

Speed Targets

Simple arithmetic/geometric: 15-20 sec
Difference-based: 30-45 sec
Complex/wrong number: 45-60 sec
If stuck > 60 sec: mark and move on

Back to 8.24 Series README