Two Arithmetic Sequences Zipped (Interleaved / Alternating)
Core Pattern: Two independent arithmetic sequences are interwoven into alternate odd and even positions. Look at every second number (stride-2) or detect alternating jump deltas (+d₁, +d₂).
Mathematical Formulation & Recurrence
\(x_{2k-1} = x_1 + (k-1)d_1 \quad \text{and} \quad x_{2k} = x_2 + (k-1)d_2\)
Method of Differences & Calculation Technique
The Stride-2 Difference Method: Instead of taking differences between immediately adjacent numbers, calculate the difference between terms separated by one number: \(\Delta_2 = x_{k+2} - x_k\). This splits the interleaved sequence into two clean parallel series: one at odd indices (positions 1, 3, 5, ...) with step \(d_1\), and one at even indices (positions 2, 4, 6, ...) with step \(d_2\). Identify which sub-sequence the question mark belongs to and add its corresponding delta.
Visual Sequence & Difference Mapping
How to Solve This Series in 3 Steps:
- 1 Separate into two tracks: Odd positions (1st, 3rd, 5th, ...) form Sequence A; even positions (2nd, 4th, 6th, ...) form Sequence B.
- 2 Compute stride-2 differences: Find d₁ for Sequence A and d₂ for Sequence B.
- 3 Identify target slot parity: If the missing term is at an odd position, continue Sequence A. If it is at an even position, continue Sequence B.
Step-by-Step Worked Examples:
3, 20, 8, 30, 13, ?
Example 1
1. Track 1 (odds): 3, 8, 13 (step +5)
2. Track 2 (evens): 20, 30, ? (step +10)
3. Pos 6 is even: 30 + 10 = 40
Answer: 40
50, 4, 45, 8, 40, 12, ?
Example 2
1. Track 1 (odds): 50, 45, 40, ? (step -5)
2. Track 2 (evens): 4, 8, 12 (step +4)
3. Pos 7 is odd: 40 - 5 = 35
Answer: 35
1, 100, 4, 90, 7, 80, 10, ?
Example 3
1. Odds: 1, 4, 7, 10 (+3)
2. Evens: 100, 90, 80, ? (-10)
3. Pos 8 is even: 80 - 10 = 70
Answer: 70
2, 5, 6, 9, 10, 13, ?
Example 4
1. Notice adjacent: +3, +1, +3, +1, +3
2. Or Track 1 (odds): 2, 6, 10, ? (+4)
3. Pos 7 is odd: 10 + 4 = 14
Answer: 14
Interactive Comprehension Test
Instant Check
Try It Yourself Now:
4, 50, 9, 45, 14, 40, ?
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