Episode 8 — Aptitude and Reasoning / 8.24 — Series
8.24.c Solved Examples -- Series
Example 1: Simple Arithmetic Series
Problem: Find the next term: 7, 13, 19, 25, 31, ?
Solution:
Series: 7, 13, 19, 25, 31, ?
D1: 6, 6, 6, 6
Constant difference = 6.
Next term = 31 + 6 = 37
Answer: 37
Example 2: Geometric Series
Problem: Find the next term: 5, 15, 45, 135, ?
Solution:
Series: 5, 15, 45, 135, ?
Ratios: x3, x3, x3
Constant ratio = 3.
Next term = 135 x 3 = 405
Answer: 405
Example 3: Difference of Differences
Problem: Find the next term: 1, 3, 7, 13, 21, 31, ?
Solution:
Series: 1, 3, 7, 13, 21, 31, ?
D1: 2, 4, 6, 8, 10
D2: 2, 2, 2, 2
D2 is constant (2).
Next D1 = 10 + 2 = 12
Next term = 31 + 12 = 43
Answer: 43
Example 4: Square-Based Series
Problem: Find the next term: 2, 5, 10, 17, 26, ?
Solution:
Check against n^2 + 1:
n=1: 1+1 = 2 (matches)
n=2: 4+1 = 5 (matches)
n=3: 9+1 = 10 (matches)
n=4: 16+1 = 17 (matches)
n=5: 25+1 = 26 (matches)
n=6: 36+1 = 37
Answer: 37
Example 5: Cube-Based Series
Problem: Find the next term: 0, 7, 26, 63, 124, ?
Solution:
Check against n^3 - 1:
n=1: 1-1 = 0 (matches)
n=2: 8-1 = 7 (matches)
n=3: 27-1 = 26 (matches)
n=4: 64-1 = 63 (matches)
n=5: 125-1 = 124 (matches)
n=6: 216-1 = 215
Answer: 215
Example 6: Fibonacci-Like Series
Problem: Find the next term: 2, 5, 7, 12, 19, 31, ?
Solution:
Check: a_n = a_(n-1) + a_(n-2)?
2 + 5 = 7 (matches)
5 + 7 = 12 (matches)
7 + 12 = 19 (matches)
12 + 19 = 31 (matches)
19 + 31 = 50
Answer: 50
Example 7: Increasing Multiplier
Problem: Find the next term: 1, 2, 6, 24, 120, ?
Solution:
Series: 1, 2, 6, 24, 120, ?
Ratios: x2, x3, x4, x5
Pattern: Multiply by increasing integers.
Next ratio = x6.
Next term = 120 x 6 = 720
(This is the factorial series: 1!, 2!, 3!, 4!, 5!, 6! = 720)
Answer: 720
Example 8: Prime Number Differences
Problem: Find the next term: 1, 3, 6, 11, 18, 29, ?
Solution:
Series: 1, 3, 6, 11, 18, 29, ?
D1: 2, 3, 5, 7, 11
D1 = 2, 3, 5, 7, 11 -> These are prime numbers!
Next prime = 13.
Next term = 29 + 13 = 42
Answer: 42
Example 9: Alternate Series
Problem: Find the next term: 3, 4, 9, 16, 27, 64, ?
Solution:
Split into odd and even positions:
Odd positions: 3, 9, 27, ? -> x3, x3, x3 -> 81
Even positions: 4, 16, 64 -> x4, x4 -> (256 next)
The 7th term is an odd position.
Answer: 81
Example 10: Wrong Number Detection
Problem: Find the wrong number: 2, 5, 10, 17, 23, 37
Solution:
Series: 2, 5, 10, 17, 23, 37
D1: 3, 5, 7, 6, 14
D1 should be an arithmetic sequence: 3, 5, 7, 9, 11
The 4th difference is 6 instead of 9.
This means the 5th term (23) is wrong.
It should be 17 + 9 = 26.
Verification: 2, 5, 10, 17, 26, 37
D1: 3, 5, 7, 9, 11 -- consistent!
Answer: 23 is the wrong number (should be 26)
Example 11: Mixed Operation Series
Problem: Find the next term: 2, 3, 6, 7, 14, 15, ?
Solution:
Pattern: +1, x2, +1, x2, +1, x2
2 (+1)-> 3 (x2)-> 6 (+1)-> 7 (x2)-> 14 (+1)-> 15 (x2)-> 30
Answer: 30
Example 12: n^2 + n Pattern
Problem: Find the next term: 2, 6, 12, 20, 30, ?
Solution:
Check n(n+1):
n=1: 1x2 = 2 (matches)
n=2: 2x3 = 6 (matches)
n=3: 3x4 = 12 (matches)
n=4: 4x5 = 20 (matches)
n=5: 5x6 = 30 (matches)
n=6: 6x7 = 42
Alternatively:
D1: 4, 6, 8, 10 -> arithmetic with d=2
Next D1 = 12. Next term = 30 + 12 = 42.
Answer: 42
Example 13: Decreasing Series
Problem: Find the next term: 100, 92, 86, 82, 80, ?
Solution:
Series: 100, 92, 86, 82, 80, ?
D1: -8, -6, -4, -2
D1: -8, -6, -4, -2 -> increasing by +2 each time.
Next D1 = -2 + 2 = 0.
Next term = 80 + 0 = 80.
Answer: 80
Example 14: Power Series with Changing Base
Problem: Find the next term: 1, 4, 27, 256, ?
Solution:
1 = 1^1
4 = 2^2
27 = 3^3
256 = 4^4
Next: 5^5 = 3125
Answer: 3125
Example 15: Product of Consecutive Numbers
Problem: Find the next term: 6, 30, 120, 360, ?
Solution:
6 = 1 x 2 x 3
30 = ... let me check ratios instead.
Ratios: 30/6=5, 120/30=4, 360/120=3
Pattern: x5, x4, x3, x2
Next term = 360 x 2 = 720
Answer: 720
Example 16: Squares of Primes
Problem: Find the next term: 4, 9, 25, 49, 121, ?
Solution:
4 = 2^2
9 = 3^2
25 = 5^2
49 = 7^2
121 = 11^2
These are squares of consecutive primes: 2, 3, 5, 7, 11, 13
Next: 13^2 = 169
Answer: 169
Example 17: Multiply and Add Pattern
Problem: Find the next term: 1, 2, 5, 14, 41, ?
Solution:
Pattern: each term = previous x 3 - 1
1 x 3 - 1 = 2 (matches)
2 x 3 - 1 = 5 (matches)
5 x 3 - 1 = 14 (matches)
14 x 3 - 1 = 41 (matches)
41 x 3 - 1 = 122
Answer: 122
Example 18: Two-Level Geometric
Problem: Find the next term: 2, 3, 5, 9, 17, 33, ?
Solution:
Series: 2, 3, 5, 9, 17, 33, ?
D1: 1, 2, 4, 8, 16
D1: 1, 2, 4, 8, 16 -> Geometric series (x2)!
Next D1 = 16 x 2 = 32
Next term = 33 + 32 = 65
Answer: 65
Example 19: Missing Middle Term
Problem: Find the missing term: 4, 9, 20, ?, 90, 183
Solution:
Series: 4, 9, 20, ?, 90, 183
D1: 5, 11, ?, ?, 93
D2: 6, ?, ?, ?
Let me try: pattern of x2 + 1, x2 + 2, x2 + 3, ...
4 x 2 + 1 = 9 (matches)
9 x 2 + 2 = 20 (matches)
20 x 2 + 3 = 43
43 x 2 + 4 = 90 (matches!)
90 x 2 + 3 = 183 (matches if we adjust the pattern)
Hmm, let me recheck: 90 x 2 + 3 = 183. Yes!
Pattern: x2+1, x2+2, x2+3, x2+4, x2+3? No, that's inconsistent.
Let me try differences:
Series: 4, 9, 20, 43, 90, 183
D1: 5, 11, 23, 47, 93
D2: 6, 12, 24, 46... Hmm.
D1: 5, 11, 23, 47, 93
Ratios in D1: ~2.2, ~2.09, ~2.04, ~1.98 -> approximately x2 + 1
5x2+1=11, 11x2+1=23, 23x2+1=47, 47x2-1=93... not quite clean.
Actually: D1: 5, 11, 23, 47, 93
Pattern: each D1 = prev x 2 + 1?
5x2+1=11 (yes), 11x2+1=23 (yes), 23x2+1=47 (yes), 47x2-1=93. Hmm, 47x2=94, not 93.
Let me just try: missing term = 43.
Answer: 43
Example 20: Wrong Number with Cubes
Problem: Find the wrong number: 1, 8, 27, 64, 124, 216
Solution:
Perfect cubes: 1^3=1, 2^3=8, 3^3=27, 4^3=64, 5^3=125, 6^3=216
The 5th term is 124 but should be 125.
Wrong number = 124.
Answer: 124 is wrong (should be 125)
Example 21: Sum of Digits Pattern
Problem: Find the next term: 10, 12, 15, 19, 14, ?
Solution:
Wait, this doesn't look like a standard arithmetic or geometric series. Let me check:
D1: 2, 3, 4, -5 -> not consistent.
Let me look for another pattern:
10: sum of digits = 1+0 = 1
12: sum of digits = 1+2 = 3
15: sum of digits = 1+5 = 6
19: sum of digits = 1+9 = 10
14: sum of digits = 1+4 = 5
Hmm, or maybe:
10, 12, 15, 19, 24, ?
D1: 2, 3, 4, 5 -> arithmetic!
Next: 24 + 6 = 30.
But the given series has 14, not 24. If it's "find the wrong number":
14 is wrong, should be 24.
Actually, let me reconsider the problem as a straightforward series:
Series: 10, 12, 15, 19, 24, ?
D1: 2, 3, 4, 5, ?
Next D1 = 6. Next term = 24 + 6 = 30.
Answer: 30 (if 14 was a typo for 24)
Example 22: Triangular Numbers
Problem: Find the next term: 1, 3, 6, 10, 15, 21, ?
Solution:
These are triangular numbers: T_n = n(n+1)/2
T_1 = 1, T_2 = 3, T_3 = 6, T_4 = 10, T_5 = 15, T_6 = 21
T_7 = 7 x 8 / 2 = 28
Alternatively:
D1: 2, 3, 4, 5, 6 -> Next D1 = 7
Next term = 21 + 7 = 28
Answer: 28
Example 23: Alternating Add and Subtract
Problem: Find the next term: 3, 5, 2, 8, 1, 11, ?
Solution:
Odd positions: 3, 2, 1, ? -> -1 each time -> 0
Even positions: 5, 8, 11 -> +3 each time -> 14
7th term (odd position) = 0
Answer: 0
Example 24: Square Root Pattern
Problem: Find the next term: 1, 1, 2, 6, 24, 120, 720, ?
Solution:
This is the factorial series!
0! or 1! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
Answer: 5040
Example 25: Complex Pattern
Problem: Find the next term: 4, 5, 9, 18, 34, ?
Solution:
Series: 4, 5, 9, 18, 34, ?
D1: 1, 4, 9, 16
D1: 1, 4, 9, 16 = 1^2, 2^2, 3^2, 4^2 (perfect squares!)
Next D1 = 5^2 = 25
Next term = 34 + 25 = 59
Answer: 59
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