Episode 8 — Aptitude and Reasoning / 8.24 — Series
8.24.b Tips, Tricks, and Shortcuts -- Series
Tip 1: The Difference Method (Most Important Technique)
Step-by-step:
- Write down the series.
- Calculate the first-level differences (D1).
- If D1 is constant -> Arithmetic series (done).
- If D1 is not constant -> Calculate second-level differences (D2).
- If D2 is constant -> Quadratic pattern.
- Continue until you find a constant level or a recognizable pattern.
Template:
Series: a1 a2 a3 a4 a5 a6
D1: d1 d2 d3 d4 d5
D2: e1 e2 e3 e4
D3: f1 f2 f3
Example:
Series: 1 3 7 13 21 31 ?
D1: 2 4 6 8 10 -> Pattern! (+2 each time)
D2: 2 2 2 2 -> Constant!
Next D1 = 10 + 2 = 12
Next term = 31 + 12 = 43
Tip 2: The Ratio Method (For Geometric/Multiplicative Series)
When differences don't show a pattern, try dividing consecutive terms:
Series: 3 12 48 192 ?
Ratios: x4 x4 x4
Next term = 192 x 4 = 768
If ratios themselves form a pattern:
Series: 2 4 12 48 240 ?
Ratios: x2 x3 x4 x5
Next ratio = x6
Next term = 240 x 6 = 1440
Tip 3: Recognizing Square and Cube Patterns
Quick Check: Is the series close to perfect squares or cubes?
If terms are: 2, 5, 10, 17, 26, 37
Compare with: 1, 4, 9, 16, 25, 36 (perfect squares)
Difference: 1, 1, 1, 1, 1, 1 -> Series = n^2 + 1
If terms are: 0, 7, 26, 63, 124
Compare with: 1, 8, 27, 64, 125 (perfect cubes)
Difference: 1, 1, 1, 1, 1 -> Series = n^3 - 1
Common Square/Cube Patterns:
n^2: 1, 4, 9, 16, 25, 36, ...
n^2 + 1: 2, 5, 10, 17, 26, 37, ...
n^2 - 1: 0, 3, 8, 15, 24, 35, ...
n^2 + n: 2, 6, 12, 20, 30, 42, ... [= n(n+1)]
n^3: 1, 8, 27, 64, 125, 216, ...
n^3 + 1: 2, 9, 28, 65, 126, 217, ...
n^3 - 1: 0, 7, 26, 63, 124, 215, ...
Tip 4: Splitting Alternate Series
If the series seems chaotic, try splitting into odd-position and even-position terms.
Series: 2, 3, 8, 9, 32, 27, ?
Odd positions: 2, 8, 32, ? -> x4 each time -> 128
Even positions: 3, 9, 27 -> x3 each time
Answer: 128
Also try splitting into groups of 3:
Series: 1, 2, 3, 4, 8, 12, 7, 32, ?
Groups of 3: (1,2,3), (4,8,12), (7,32,?)
First of each: 1, 4, 7 -> +3
Second of each: 2, 8, 32 -> x4
Third of each: 3, 12, ? -> x4 -> 48
Tip 5: Wrong Number Detection Strategy
Step 1: Identify the expected pattern.
Step 2: Compute what each term SHOULD be.
Step 3: The term that doesn't fit is wrong.
Quick Trick:
If 5 out of 6 numbers follow a pattern and 1 doesn't, that 1 is the wrong number.
Series: 2, 5, 10, 17, 23, 37
Pattern: n^2 + 1 -> 2, 5, 10, 17, 26, 37
23 should be 26. Wrong number = 23.
Alternative: Differences method
Series: 2, 5, 10, 17, 23, 37
D1: 3, 5, 7, 6, 14 <- 6 breaks the pattern (should be 9)
If D1 should be 3, 5, 7, 9, 11:
Fix: 17 + 9 = 26 (not 23). So 23 is wrong.
Tip 6: Common Multiplicative Patterns
| Pattern | Example | Description |
|---|---|---|
| x1, x2, x3, x4... | 1, 1, 2, 6, 24, 120 | Factorial-like |
| x2, x2, x2... | 3, 6, 12, 24, 48 | Doubling |
| x1.5, x1.5... | 4, 6, 9, 13.5 | Constant ratio 1.5 |
| x2, x3, x4... | 1, 2, 6, 24, 120 | Increasing multiplier |
| x(-1), x(-2)... | 2, -2, 4, -8, 16 | Alternating sign |
Tip 7: Recognizing Fibonacci-Like Patterns
Classic Fibonacci: a_n = a_(n-1) + a_(n-2)
1, 1, 2, 3, 5, 8, 13, 21, ...
Check: 1+1=2, 1+2=3, 2+3=5, 3+5=8, ...
Quick Test:
For any three consecutive terms a, b, c: check if a + b = c.
Series: 4, 7, 11, 18, 29, ?
Check: 4+7=11, 7+11=18, 11+18=29 -> Yes! Fibonacci-like.
Next: 18+29 = 47
Variations:
Sum of 3: a_n = a_(n-1) + a_(n-2) + a_(n-3)
Product: a_n = a_(n-1) * a_(n-2)
Subtract: a_n = a_(n-1) - a_(n-2)
Tip 8: Pattern Recognition by Term Values
If terms are small (1-50): Think arithmetic, simple patterns.
If terms grow rapidly (doubling, tripling): Think geometric or power series.
If terms include 1, 4, 9, 16, 25: Think squares.
If terms include 1, 8, 27, 64: Think cubes.
If terms include 2, 3, 5, 7, 11: Think primes.
If terms are very large (1000+): Think cubes or high powers.
Tip 9: The "Near-Square/Cube" Technique
When terms are close to perfect squares or cubes, check:
Terms: 3, 10, 29, 66, 127
Cubes: 1, 8, 27, 64, 125
Diff: 2, 2, 2, 2, 2 -> Terms = n^3 + 2
Terms: 2, 6, 12, 20, 30
Squares: 1, 4, 9, 16, 25
Hmm, not directly. But:
2=1x2, 6=2x3, 12=3x4, 20=4x5, 30=5x6 -> n(n+1)
Tip 10: Mixed Operation Pattern Recognition
Some series alternate between two operations:
Series: 5, 10, 8, 16, 14, 28, 26, ?
Pattern: x2, -2, x2, -2, x2, -2, ?
5 x2=10, 10-2=8, 8x2=16, 16-2=14, 14x2=28, 28-2=26, 26x2=52
Answer: 52
Series: 3, 4, 8, 9, 27, 28, ?
Pattern: +1, x2, +1, x3, +1, x4
3+1=4, 4x2=8, 8+1=9, 9x3=27, 27+1=28, 28x4=112? Hmm.
Or: +1 alternates with some multiplication.
Odd positions: 3, 8, 27, ? -> 3=3^1, 8=2^3, 27=3^3 -> powers pattern
Even positions: 4, 9, 28 -> 4=3+1, 9=8+1, 28=27+1 -> previous+1
Next odd = 4^3 = 64 (if pattern is n^3). Then even = 65.
Answer: 64 (if asking for position 7)
Tip 11: Letter/Alphabet Series (Bonus)
Position Values: A=1, B=2, C=3, ..., Z=26
Common Patterns:
A, C, E, G, I, ... -> Skip 1 letter (odd positions)
A, D, G, J, M, ... -> Skip 2 letters (+3 each)
B, D, H, P, ... -> Positions: 2, 4, 8, 16 (doubling)
Z, X, V, T, R, ... -> Reverse skip 1
Alpha-Numeric Mixed:
A2, D5, G8, J11, ? -> Letters: A,D,G,J (+3) -> M
-> Numbers: 2,5,8,11 (+3) -> 14
-> Answer: M14
Tip 12: Speed Strategy for Exams
Time Allocation:
- Simple arithmetic/geometric: 15-20 seconds
- Difference-based: 30-45 seconds
- Complex (wrong number, mixed): 45-60 seconds
- If unsure after 60 seconds: Mark and move on
Quick Decision Tree:
1. Glance at the series.
2. Are terms growing slowly? -> Check differences (arithmetic)
3. Are terms growing fast? -> Check ratios (geometric)
4. Are terms near squares/cubes? -> Check n^2 +/- k or n^3 +/- k
5. Does the series look chaotic? -> Split odd/even positions
6. Is it "find the wrong number"? -> Find the pattern, spot the outlier
7. None of the above? -> Try Fibonacci, primes, mixed ops
Tip 13: Common Traps in Series Problems
Trap 1: Assuming constant difference when it's not
- Always verify with at LEAST 3 differences before concluding.
Trap 2: Confusing n^2 + 1 with (n+1)^2
- 2, 5, 10, 17 is n^2 + 1 (for n = 1,2,3,4)
- NOT 2^2, ... (that gives 4, not 5)
Trap 3: In "wrong number" problems, the wrong number may be anywhere
- Don't assume it's the last term.
- Check the pattern against ALL terms.
Trap 4: Multiple valid patterns
- Sometimes two patterns seem to fit. Use more terms to disambiguate.
- In exams, the "simpler" pattern is usually the intended one.
Previous: 8.24.a Concepts and Formulas | Next: 8.24.c Solved Examples