Number Series Concepts

1. Arithmetic & Geometric Progressions

• Idea: Sequences driven by constant differences or multiplying factors.
• Rule: AP: each term = previous term + constant d. GP: each term = previous term * constant r. nth term = a + (n-1)d for AP, a * r^(n-1) for GP.
• Example:
• Problem: Find the next term in the series 3, 7, 11, 15, 19.
• Solution:
• Step 1: Subtract each term from the next: 7-3 = 4, 11-7 = 4, 15-11 = 4, 19-15 = 4. The difference between consecutive terms is constant at +4, so this is an arithmetic progression.
• Step 2: Add the common difference to the last term: 19 + 4 = 23.
• Answer: 23

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• Problem: Find the next term in the series 2, 6, 18, 54.
• Solution:
• Step 1: Divide each term by the previous: 6/2 = 3, 18/6 = 3, 54/18 = 3. Each term is multiplied by a constant factor of 3, so this is a geometric progression.
• Step 2: Multiply the last term by the common ratio: 54 * 3 = 162.
• Answer: 162

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2. Difference of Differences (Two-Tier)

• Idea: Finding patterns in secondary or tertiary level differences.
• Rule: Compute first differences, then second differences if needed. If second diff is constant, series follows a quadratic pattern.
• Example:
• Problem: Find the next term in the series 2, 5, 10, 17, 26.
• Solution:
• Step 1: Compute first differences between consecutive terms: 5-2 = 3, 10-5 = 5, 17-10 = 7, 26-17 = 9. The first differences are 3, 5, 7, 9.
• Step 2: Compute second differences: 5-3 = 2, 7-5 = 2, 9-7 = 2. The second difference is constant at +2, meaning the series follows a quadratic pattern like n^2 + 1.
• Step 3: The next first difference will be 9 + 2 = 11. Add it to the last term: 26 + 11 = 37.
• Answer: 37

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3. Square and Cube Variations

• Idea: Series built around n^2 +/- 1, n^3 +/- k, or prime offsets.
• Rule: Identify nearest square/cube and note the constant offset from it.
• Example:
• Problem: Find the next term in the series 2, 5, 10, 17, 26, 37.
• Solution:
• Step 1: Compare each term to perfect squares: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36.
• Step 2: Notice every term is exactly one more than a square: 1+1 = 2, 4+1 = 5, 9+1 = 10, 16+1 = 17, 25+1 = 26, 36+1 = 37. The rule is n^2 + 1.
• Step 3: The 7th term is 7^2 + 1 = 49 + 1 = 50.
• Answer: 50

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• Problem: Find the next term in the series 0, 7, 26, 63, 124.
• Solution:
• Step 1: Compare each term to perfect cubes: 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125.
• Step 2: Notice every term is exactly one less than a cube: 1-1 = 0, 8-1 = 7, 27-1 = 26, 64-1 = 63, 125-1 = 124. The rule is n^3 - 1.
• Step 3: The 6th term is 6^3 - 1 = 216 - 1 = 215.
• Answer: 215

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4. Alternate / Twin Series

• Idea: Interleaved sequences where two alternating patterns run simultaneously.
• Rule: Separate terms at odd positions and even positions into two different patterns.
• Example:
• Problem: Find the next term in the series 3, 10, 6, 15, 9, 20, 12, ?.
• Solution:
• Step 1: Split the series by position. Odd positions (1st, 3rd, 5th, 7th): 3, 6, 9, 12. Each increases by +3.
• Step 2: Even positions (2nd, 4th, 6th): 10, 15, 20. Each increases by +5.
• Step 3: The missing term is at position 8 (even), so use the even pattern: 20 + 5 = 25.
• Answer: 25

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5. Fibonacci & Recursive Patterns

• Idea: Sequences where terms are generated by adding previous elements.
• Rule: Standard Fibonacci: T(n) = T(n-1) + T(n-2). Some series use three-term sums or variations.
• Example:
• Problem: Find the next term in the series 2, 3, 5, 8, 13, 21.
• Solution:
• Step 1: Check if each term is the sum of the two before it: 2+3 = 5, 3+5 = 8, 5+8 = 13, 8+13 = 21. The pattern holds for all terms.
• Step 2: Sum the last two terms: 13 + 21 = 34.
• Answer: 34

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• Problem: Find the next term in the series 1, 2, 3, 6, 11, 20.
• Solution:
• Step 1: Check if each term is the sum of the three before it: 1+2+3 = 6, 2+3+6 = 11, 3+6+11 = 20. The pattern holds.
• Step 2: Sum the last three terms: 6 + 11 + 20 = 37.
• Answer: 37

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6. Mixed Operations (a*x +/- b)

• Idea: Complex step patterns combining multiplication/division with addition/subtraction.
• Rule: Pattern may be like: multiply by k then add c, or follow a stepwise formula T(n+1) = a * T(n) + b.
• Example:
• Problem: Find the next term in the series 3, 7, 15, 31, 63.
• Solution:
• Step 1: Look for a consistent operation linking consecutive terms. Try multiply by 2 and add 1: 32+1 = 7, 72+1 = 15, 152+1 = 31, 312+1 = 63. Each term is double the previous plus 1.
• Step 2: Apply the same to the last term: 63 * 2 + 1 = 127.
• Answer: 127

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• Problem: Find the next term in the series 100, 96, 104, 100, 108.
• Solution:
• Step 1: Notice the operations alternate: 100-4 = 96, 96+8 = 104, 104-4 = 100, 100+8 = 108. The pattern cycles between subtract 4 and add 8.
• Step 2: The last operation was add 8, so the next is subtract 4: 108 - 4 = 104.
• Answer: 104

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7. Prime Number Series

• Idea: Progressions tracking sequential or gap-skipped prime numbers.
• Rule: Identify primes in order. Sometimes series includes twin primes, primes at specific gaps, or prime-related composites.
• Example:
• Problem: Find the next term in the series 2, 3, 5, 7, 11, 13.
• Solution:
• Step 1: Identify these as consecutive prime numbers (numbers divisible only by 1 and themselves).
• Step 2: The next prime after 13 is 17.
• Answer: 17

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• Problem: Find the next term in the series 7, 11, 13, 17, 19.
• Solution:
• Step 1: Recognize these as primes between 7 and 20.
• Step 2: The next prime after 19 is 23.
• Answer: 23

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• Problem: Find the next term in the series 4, 6, 10, 12, 16.
• Solution:
• Step 1: Recognize each term as a prime number minus 1: 5-1 = 4, 7-1 = 6, 11-1 = 10, 13-1 = 12, 17-1 = 16.
• Step 2: The next prime after 17 is 19, so the next term is 19 - 1 = 18.
• Answer: 18

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8. Factorial Series

• Idea: Sequences built on ascending or descending factorial computations.
• Rule: n! = n * (n-1) * (n-2) * … * 1. Series may use n! alone or with offsets.
• Example:
• Problem: Find the next term in the series 1, 2, 6, 24, 120.
• Solution:
• Step 1: Recognize these as factorials: 1! = 1, 2! = 21 = 2, 3! = 321 = 6, 4! = 4321 = 24, 5! = 54321 = 120.
• Step 2: The next term is 6! = 65432*1 = 720.
• Answer: 720

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• Problem: Find the next term in the series 2, 3, 7, 25, 121.
• Solution:
• Step 1: Compare each term to factorial values: 1!+1 = 1+1 = 2, 2!+1 = 2+1 = 3, 3!+1 = 6+1 = 7, 4!+1 = 24+1 = 25, 5!+1 = 120+1 = 121.
• Step 2: The pattern is n! + 1. The 6th term is 6! + 1 = 720 + 1 = 721.
• Answer: 721

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9. Triangular & Polygonal Number Series

• Idea: Sequences based on dot-pattern geometric growth numbers.
• Rule: Triangular numbers: T(n) = n(n+1)/2. Square: n^2. Pentagonal: n(3n-1)/2. Hexagonal: n(2n-1).
• Example:
• Problem: Find the next term in the series 1, 3, 6, 10, 15.
• Solution:
• Step 1: Recognize these as triangular numbers: T(1) = 12/2 = 1, T(2) = 23/2 = 3, T(3) = 34/2 = 6, T(4) = 45/2 = 10, T(5) = 5*6/2 = 15. Each triangular number is the sum of natural numbers up to n.
• Step 2: The next term is T(6) = 6*7/2 = 42/2 = 21.
• Answer: 21

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• Problem: Find the next term in the series 1, 4, 9, 16, 25.
• Solution:
• Step 1: Recognize these as perfect squares: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25.
• Step 2: The next term is 6^2 = 36.
• Answer: 36

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10. Alternating Operation Series

• Idea: Cycles shifting dynamically between addition, multiplication, subtraction, and division.
• Rule: The operation changes at each step following a fixed cycle or pattern.
• Example:
• Problem: Find the next term in the series 10, 12, 36, 33, 99, 95.
• Solution:
• Step 1: Examine operations between consecutive terms: 10+2 = 12, 123 = 36, 36-3 = 33, 333 = 99, 99-4 = 95. The pattern alternates between adding/subtracting (with values 2, 3, 4…) and multiplying by 3.
• Step 2: The last operation was subtract 4, so the next operation is multiply by 3: 95 * 3 = 285.
• Answer: 285

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11. Missing Term vs. Next Term

• Idea: Solving for gaps embedded within the middle of a sequence rather than at the end.
• Rule: Find the pattern using the terms before and after the gap, then verify consistency.
• Example:
• Problem: Find the missing term in the series 2, 5, ?, 17, 26.
• Solution:
• Step 1: Look for a pattern in the known terms. Compare to squares: 1^2+1 = 2, 2^2+1 = 5, 4^2+1 = 17, 5^2+1 = 26. The pattern is n^2 + 1.
• Step 2: The missing term is at position 3: 3^2 + 1 = 9 + 1 = 10.
• Step 3: Verify consistency: first differences would be 3, 5, 7, 9 (all increasing by +2). The second difference is constant at +2, confirming the pattern.
• Answer: 10

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• Problem: Find the missing term in the series 3, 7, ?, 31, 63.
• Solution:
• Step 1: Look for a pattern: 32+1 = 7, 152+1 = 31, 31*2+1 = 63. Each term is double the previous plus 1.
• Step 2: The missing term is what follows 7 in this pattern: 7*2+1 = 15.
• Step 3: Verify: 15*2+1 = 31, which matches the next given term. The pattern is consistent.
• Answer: 15

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12. Wrong Term Identification

• Idea: Spotting the single outlier digit that breaks a complex sequence rule.
• Rule: Test the pattern from the first two terms, then check each subsequent term. The term that doesn’t follow the rule is the wrong one.
• Example:
• Problem: Identify the wrong term in the series 2, 6, 12, 20, 28, 42.
• Solution:
• Step 1: Check the pattern n(n+1): 12 = 2, 23 = 6, 34 = 12, 45 = 20, 56 = 30 (not 28), 67 = 42.
• Step 2: The term 28 breaks the pattern and should be 5*6 = 30.
• Answer: 28 (should be 30)

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• Problem: Identify the wrong term in the series 1, 4, 9, 16, 24, 36.
• Solution:
• Step 1: Check the pattern n^2: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25 (not 24), 6^2 = 36.
• Step 2: The term 24 breaks the pattern and should be 5^2 = 25.
• Answer: 24 (should be 25)