Nature's Favourite Sequence

Class 7 Maths · Chapter 6: Number Play · NCERT Ganita Prakash section 6.4

Learn

In Sanskrit, Prakrit and many Indian languages, poetry has short syllables (1 beat) and long syllables (2 beats). Poets asked: how many rhythms fill exactly n beats? This is the same as writing n as a sum of 1s and 2s: 3 = 1 + 1 + 1 = 1 + 2 = 2 + 1, so 3 ways.

n = 1, 2, 3, 4: 1, 2, 3, 5 ways

The clever method: every 5-beat rhythm starts with 1 + (a 4-beat rhythm) or 2 + (a 3-beat rhythm). So ways(5) = ways(4) + ways(3) = 5 + 3 = 8, and ways(6) = 8 + 5 = 13. Each number is the sum of the two before: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …

8 = 5 + 3 · 13 = 8 + 5

The Prakrit scholar Virahāṅka gave this rule around 700 CE, inspired by Piṅgala (around 300 BCE); Gopala and Hemachandra wrote about it later. In the West they are called Fibonacci numbers, after a writer of 1202 CE, about 500 years later. So we say Virahāṅka–Fibonacci numbers.

Patterns: the parities go odd, even, odd, odd, even, odd, odd, even … repeating every 3, so the 20th number is even (20 = 3 × 6 + 2, like position 2). In nature, daisies often have 13, 21 or 34 petals. And a staircase of 8 steps climbed 1 or 2 at a time has 34 ways!

O, E, O, O, E, O, O, E …

Explore: all the ways to make 4.

1. Tap every way of writing 4 as a sum of 1s and 2s.

Remember

• The number of rhythms of n beats (sums of 1s and 2s) is the nth Virahāṅka number.
• Each Virahāṅka number is the sum of the two before it: 1, 2, 3, 5, 8, 13 …

Quick check

Two Virahāṅka numbers in a row are 987 and 1597. What comes next?

1. 2584
2. 2207
3. 610
4. 2594

Think about it

How many ways are there to write 7 as a sum of 1s and 2s?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.