Recursive Sequence (Fibonacci & Linear Recurrence)
Core Pattern: In a recursive sequence, each term depends directly on the preceding two terms. In Fibonacci-type series, every number is the exact sum of the two numbers before it.
Mathematical Formulation & Recurrence
\(x_n = x_{n-1} + x_{n-2} \quad \text{or} \quad x_n = a \cdot x_{n-1} + b \cdot x_{n-2}\)
Method of Differences & Calculation Technique
Difference Signature of Recurrence: If you take differences on a Fibonacci sequence (e.g. 1, 1, 2, 3, 5, 8, 13 \(\to\) \(\Delta = 0, 1, 1, 2, 3, 5\)), you discover an astounding signature: \(x_k - x_{k-1} = x_{k-2}\)! The differences match the sequence itself shifted by one position. This reveals that addition is at play, but across consecutive terms rather than a constant. Formula test: Check if \(x_{k-2} + x_{k-1} = x_k\). Next term = \(x_{\text{last}} + x_{\text{second-last}}\).
Visual Sequence & Difference Mapping
How to Solve This Series in 3 Steps:
- 1 Look for the sum of neighbors: Check if term 3 = term 1 + term 2 (e.g. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8).
- 2 Check generalized Lucas/Fibonacci seeds: Sequences can start from any numbers (e.g. 2, 5, 7, 12, 19, 31).
- 3 Check weighted recurrence: If simple sum is too small, check a·x(n-1) + b·x(n-2) (e.g. 2×prev + prev2).
Step-by-Step Worked Examples:
1, 1, 2, 3, 5, 8, 13, ?
Example 1
1. Check sums: 1+1=2, 1+2=3, 2+3=5, 3+5=8
2. Fibonacci rule: x(n) = x(n-1) + x(n-2)
3. Next term: 8 + 13 = 21
Answer: 21
2, 5, 7, 12, 19, 31, ?
Example 2
1. 2+5=7, 5+7=12, 7+12=19, 12+19=31
2. Sum of previous two terms
3. Next term: 19 + 31 = 50
Answer: 50
3, 4, 7, 11, 18, 29, ?
Example 3
1. Lucas-style: 3+4=7, 4+7=11, 7+11=18
2. Sum of last two terms
3. Next term: 18 + 29 = 47
Answer: 47
1, 2, 5, 12, 29, ? (Pell sequence)
Example 4
1. Check: 2×2 + 1 = 5; 2×5 + 2 = 12; 2×12 + 5 = 29
2. Recurrence: x(n) = 2·x(n-1) + x(n-2)
3. Next term: 2×29 + 12 = 70
Answer: 70
Interactive Comprehension Test
Instant Check
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4, 6, 10, 16, 26, 42, ?
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