Concept 8 of 19Foundation
Watch on YouTubeVideoAlternating & repeating patterns
A pattern can repeat a short block forever. Its length is called the period.
- A, B, A, B, A, B, … (period 2)
- A, B, C, D, A, B, C, D, … (period 4)
- Red, Green, Blue, Red, Green, Blue, … (period 3)
To find the nth item: compute n ÷ period. The remainder tells you which item in the block.
(If remainder = 0, the last item of the block is used.)
Example
Block "ABCD" (period 4). What's the 100th letter?
100 ÷ 4 = 25 remainder 0 → use LAST in block → D.
What's the 27th? 27 ÷ 4 = 6 rem 3 → 3rd in block → C.
100 ÷ 4 = 25 remainder 0 → use LAST in block → D.
What's the 27th? 27 ÷ 4 = 6 rem 3 → 3rd in block → C.
💡 Tip:When the remainder is 0, use the last item — don't use the first!
▸Why does this work? (derivation)
Why remainder? After each full block of "period" items, the pattern restarts. So position n matches position n mod (period). We just need to know where we are within the current cycle.
🎯 Try it!
5 questions to check what you just read.
0 / 5
- Q1.100th letter in ABABAB… ?
- Q2.50th letter in ABCDABCDABCD… ?
- Q3.Period of 3, 7, 2, 3, 7, 2, 3, 7, 2?
- Q4.15th symbol in 🔴 🟡 🔵 🔴 🟡 🔵 …?
- Q5.Day 100 days after Monday (7-day cycle)?