Episode 8 — Aptitude and Reasoning / 8.24 — Series
8.24.a Concepts and Formulas -- Series
1. What is a Number Series?
A number series is a sequence of numbers that follow a particular pattern or rule. The task is to:
- Find the next number in the series, OR
- Find a missing number in the series, OR
- Identify the wrong number that breaks the pattern.
2. Types of Number Series
Type 1: Arithmetic Series (Constant Difference)
Each term differs from the previous by a constant value (d).
Formula: a_n = a_1 + (n-1) * d
Example: 3, 7, 11, 15, 19, ?
Difference: +4, +4, +4, +4
Next term: 19 + 4 = 23
Example: 50, 43, 36, 29, ?
Difference: -7, -7, -7
Next term: 29 - 7 = 22
Type 2: Geometric Series (Constant Ratio)
Each term is obtained by multiplying the previous term by a constant ratio (r).
Formula: a_n = a_1 * r^(n-1)
Example: 2, 6, 18, 54, ?
Ratio: x3, x3, x3
Next term: 54 x 3 = 162
Example: 1000, 200, 40, 8, ?
Ratio: /5, /5, /5
Next term: 8 / 5 = 1.6
Type 3: Difference Series (Differences Form a Pattern)
The differences between consecutive terms themselves form a series.
Level 1 Difference (Arithmetic differences):
Series: 2, 5, 10, 17, 26, ?
Diff (L1): 3, 5, 7, 9, ?
Diff (L2): 2, 2, 2
L1 is an arithmetic series with d=2.
Next L1 diff = 9 + 2 = 11
Next term = 26 + 11 = 37
Level 2 Difference:
Series: 1, 2, 6, 15, 31, 56, ?
Diff (L1): 1, 4, 9, 16, 25
Diff (L2): 3, 5, 7, 9
Diff (L3): 2, 2, 2
L1 differences are: 1, 4, 9, 16, 25 = perfect squares!
Next L1 = 36 (6^2)
Next term = 56 + 36 = 92
Type 4: Multiplication/Division Series
Each term is obtained by multiplying or dividing by a changing value.
Example: 2, 6, 30, 210, ?
Pattern: x3, x5, x7, x9
Next term: 210 x 9 = 1890
Example: 3, 6, 18, 72, 360, ?
Pattern: x2, x3, x4, x5, x6
Next term: 360 x 6 = 2160
Type 5: Square-Based Series
Terms are related to perfect squares.
Type 5a: Perfect squares
1, 4, 9, 16, 25, 36, ... = 1^2, 2^2, 3^2, 4^2, 5^2, 6^2
Type 5b: n^2 + constant
2, 5, 10, 17, 26, ... = 1^2+1, 2^2+1, 3^2+1, 4^2+1, 5^2+1
Type 5c: n^2 - constant
0, 3, 8, 15, 24, ... = 1^2-1, 2^2-1, 3^2-1, 4^2-1, 5^2-1
Type 5d: Squares of specific sequence
4, 9, 25, 49, 121, ... = 2^2, 3^2, 5^2, 7^2, 11^2 (squares of primes)
Type 6: Cube-Based Series
Terms are related to perfect cubes.
Type 6a: Perfect cubes
1, 8, 27, 64, 125, ... = 1^3, 2^3, 3^3, 4^3, 5^3
Type 6b: n^3 + constant
2, 9, 28, 65, 126, ... = 1^3+1, 2^3+1, 3^3+1, 4^3+1, 5^3+1
Type 6c: n^3 - constant
0, 7, 26, 63, 124, ... = 1^3-1, 2^3-1, 3^3-1, 4^3-1, 5^3-1
Type 6d: Difference of cubes
7, 19, 37, 61, 91, ...
Differences: 12, 18, 24, 30 (arithmetic, d=6)
These are related to n^3 - (n-1)^3 = 3n^2 - 3n + 1
Type 7: Fibonacci-Like Series
Each term is the sum of the two preceding terms (or a variation).
Classic Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Rule: a_n = a_(n-1) + a_(n-2)
Variation 1: 2, 3, 5, 8, 13, 21, ...
Same rule, different starting values.
Variation 2 (sum of three): 1, 1, 1, 3, 5, 9, 17, ...
Rule: a_n = a_(n-1) + a_(n-2) + a_(n-3)
Variation 3 (product): 1, 2, 2, 4, 8, 32, ...
Rule: a_n = a_(n-1) * a_(n-2)
Type 8: Prime Number Series
Series involving prime numbers.
Type 8a: Prime numbers themselves
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
Type 8b: Differences are primes
1, 3, 6, 11, 18, 29, ?
Diff: 2, 3, 5, 7, 11 (primes)
Next diff = 13, Next term = 29 + 13 = 42
Type 8c: Multiplied by primes
2, 6, 30, 210, ?
Pattern: x3, x5, x7 (consecutive odd primes)
Next: 210 x 11 = 2310
Type 9: Alternate Series (Two or More Interleaved)
Two or more independent series are interleaved.
Example: 3, 5, 9, 10, 27, 15, ?
Separate into odd and even positions:
Odd positions: 3, 9, 27, ? -> x3 each time -> 81
Even positions: 5, 10, 15 -> +5 each time
Answer: 81
Example: 1, 2, 4, 8, 7, 32, 10, ?
Odd positions: 1, 4, 7, 10 -> +3 (arithmetic)
Even positions: 2, 8, 32, ? -> x4 each time -> 128
Answer: 128
Type 10: Mixed Operation Series
Different operations are applied alternately.
Example: 5, 6, 14, 15, 45, 46, ?
Pattern: +1, x2+2, +1, x3, +1, x3...
Actually: +1, then another operation alternating.
5 (+1) -> 6 (x2+2) -> 14 (+1) -> 15 (x3) -> 45 (+1) -> 46 (x?) -> ?
Better pattern: +1 alternates with x-something.
5, +1=6, 6+8=14, +1=15, 15+30=45, +1=46, 46+92=138
Or: 5, 6, 14, 15, 45, 46, 138
Type 11: Power Series
Terms follow a pattern of increasing powers.
Type 11a: Same base, increasing power
2, 4, 8, 16, 32, 64, ... = 2^1, 2^2, 2^3, 2^4, 2^5, 2^6
Type 11b: Increasing base, same power
1, 4, 9, 16, 25, ... = 1^2, 2^2, 3^2, 4^2, 5^2
Type 11c: n^n pattern
1, 4, 27, 256, 3125, ... = 1^1, 2^2, 3^3, 4^4, 5^5
Type 11d: Mixed powers
1, 8, 9, 64, 25, ... = 1^1, 2^3, 3^2, 4^3, 5^2, ...
(alternating squares and cubes)
Type 12: Wrong Number in Series
One number in the series is incorrect. Find it.
Example: 2, 5, 10, 17, 24, 37
Expected: 2, 5, 10, 17, 26, 37 (n^2 + 1)
Check: 1^2+1=2, 2^2+1=5, 3^2+1=10, 4^2+1=17, 5^2+1=26, 6^2+1=37
24 should be 26. Wrong number = 24.
Approach for wrong number:
- Find the pattern the series SHOULD follow.
- Compute what each term should be.
- The term that doesn't match is the wrong one.
Type 13: Missing Number in Series (Middle Term)
A number in the middle of the series is missing.
Example: 3, 8, ?, 24, 35
Differences: 5, ?, ?, 11
If differences are: 5, 7, 9, 11 (arithmetic, d=2)
Then: 8 + 7 = 15
Missing number = 15
3. Important Number References
Perfect Squares (Memorize 1-30)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100,
121, 144, 169, 196, 225, 256, 289, 324, 361, 400,
441, 484, 529, 576, 625, 676, 729, 784, 841, 900
Perfect Cubes (Memorize 1-15)
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,
1331, 1728, 2197, 2744, 3375
Prime Numbers (First 25)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97
Powers of 2
2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64,
2^7=128, 2^8=256, 2^9=512, 2^10=1024
Powers of 3
3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243, 3^6=729
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610