Number Play · Lesson 8 of 10
Exploring the Virahāṅka Sequence
“Extend the rhythm-count sequence and discover its parity pattern and staircase connection.”
• Find later or earlier terms using the two-term addition rule. • Use the sequence to count eight-beat rhythms and staircase paths. • Explain the repeating odd-even-even-position pattern accurately. • Predict the parity of a distant term without computing its full value. • Describe the chapter’s nature examples without treating them as universal rules.
The next term uses the previous two
The rhythm counts make the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … . Each term after the first two is the sum of the two immediately before it: for example, 21 + 34 = 55. The eighth term is 34, so there are 34 rhythms of eight beats. Position matters: the sequence begins with 1 as its first term and 2 as its second.
Problem
What follows 55, 89?
- 1.Add 55 + 89 = 144.
- 2.Then add 89 + 144 = 233.
- 3.Then add 144 + 233 = 377. The next three terms are 144, 233, 377.
Use the same rule in reverse
Knowing two adjacent terms also lets us recover an earlier term. Since earlier + later-earlier = later, subtract the smaller known term from the larger to step back. A pair of consecutive terms can therefore anchor the sequence in both directions.
Problem
Consecutive terms are 987 and 1597. Find two previous and two next terms.
- 1.Next: 987 + 1597 = 2584, then 1597 + 2584 = 4181.
- 2.Previous: 1597 − 987 = 610, then 987 − 610 = 377.
- 3.The run is 377, 610, 987, 1597, 2584, 4181.
Predict odd and even without large sums
Write only the parities of the terms: odd, even, odd, odd, even, odd, odd, even, … . Adding an odd and an even gives odd; adding two odd numbers gives even. After the first term, the three-parity pattern odd, odd, even repeats. Equivalently, positions 2, 5, 8, 11, … are even: they are positions two more than a multiple of three.
| Positions | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Parity | odd | even | odd | odd | even | odd | odd | even | odd |
Problem
Is the 20th term odd or even without calculating it?
- 1.The even terms occur at positions 2, 5, 8, 11, 14, 17, 20, … .
- 2.Position 20 belongs to that list. Therefore the 20th term is even.
The same count on a staircase
Suppose you climb eight steps, moving one or two steps at a time. Each complete route is an ordered list of 1s and 2s totaling eight. That is exactly the same counting problem as an eight-beat rhythm, so the answer is 34 routes. The sequence applies whenever the choices and the rule for combining smaller cases match this model.
Problem
How many ways can you climb five steps using moves of one or two steps?
- 1.Routes beginning with a one-step move leave four steps, which have 5 possible routes.
- 2.Routes beginning with a two-step move leave three steps, which have 3 possible routes.
- 3.Thus there are 5 + 3 = 8 routes, the fifth sequence term.
The chapter also shows daisies with 13, 21, and 34 petals and describes links between this sequence and art, science, and nature. These pictures offer examples to notice and investigate. They do not mean that every flower has a sequence-number petal count.
The eighth term is 34, not 8. Likewise, the 20th term is even because of its position in the repeating parity pattern, not because 20 itself is even.
Quiz
What is the eighth term of 1, 2, 3, 5, …?
What follows 89, 144?
What comes immediately before 987, 1597?
Which positions have even terms in the sequence?
How many one- or two-step routes climb an eight-step staircase?
Practice Problems
- Find the next three terms after 34, 55, 89. Show each addition.
- Given consecutive terms 21 and 34, find the preceding two and following two terms.
- Write the first twelve term parities and mark every even term.
- Determine the parity of the 20th term using the repeating pattern.
- Explain why staircase routes of eight steps are counted by the eighth rhythm-count term.
- What can the flower photographs illustrate, and what stronger statement would they not prove?
Key Takeaways
• Each new term is the sum of the two terms before it. • To work backward from two adjacent terms, subtract the earlier known term from the later one. • Terms in positions 2, 5, 8, 11, … are even; the 20th term is even. • One- or two-step routes and one- or two-beat rhythms follow the same counting rule. • Examples from nature invite investigation but are not universal claims.