NCERT Solutions Ganita Prakash (Part 1) Chapter 6 –141Section 6.4 Nature’s Favourite Sequence: The Virahāṅka–Fibonacci Numbers! — In-text Questions

Book page 140 Updated on2026-09-05

Q1.

How many rhythms are there with 8 beats consisting of short syllables (1 beat) and long syllables (2 beats)? Here are some possibilities: long long long long / short short short short short short short short / short long long short long / long long short short long. Can you find others?

Answer

Yes, many more. A short syllable takes 1 beat and a long syllable takes 2 beats, so every rhythm is a way of making 8 out of 1s and 2s.

Rhythm	As a sum
short short long long long	1 + 1 + 2 + 2 + 2
long short long short long	2 + 1 + 2 + 1 + 2
long long long short short	2 + 2 + 2 + 1 + 1
short long short long long	1 + 2 + 1 + 2 + 2
long short short short long long	2 + 1 + 1 + 1 + 2 + 2
short short short short long long	1 + 1 + 1 + 1 + 2 + 2

Altogether there are 34 such 8-beat rhythms.

Tip: counting them one by one is slow and easy to get wrong. The systematic method in the book gives the answer without listing.

Q2.

Phrased more mathematically: In how many different ways can one write a number, say 8, as a sum of 1’s and 2’s? For example, we have: 8 = 2 + 2 + 2 + 2, 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 8 = 1 + 2 + 2 + 1 + 2, 8 = 2 + 2 + 1 + 1 + 2, etc. Do you see other ways?

Answer

There are 34 ways. Build them up step by step instead of listing them all.

n	1	2	3	4	5	6	7	8
Number of ways	1	2	3	5	8	13	21	34

ways(3) = ways(2) + ways(1) = 2 + 1 = 3
ways(4) = ways(3) + ways(2) = 3 + 2 = 5
ways(5) = ways(4) + ways(3) = 5 + 3 = 8
ways(6) = ways(5) + ways(4) = 8 + 5 = 13
ways(7) = ways(6) + ways(5) = 13 + 8 = 21
ways(8) = ways(7) + ways(6) = 21 + 13 = 34

Some more ways for 8: 1 + 1 + 2 + 2 + 2, 2 + 1 + 2 + 1 + 2, 2 + 2 + 1 + 2 + 1, 1 + 2 + 1 + 1 + 1 + 2, 2 + 1 + 1 + 1 + 1 + 2.

Why it happens: every sum for 8 must begin with either a 1 or a 2. If it begins with 1, the rest makes 7; if it begins with 2, the rest makes 6. So the count for 8 is the count for 7 plus the count for 6.

Q3.

Try writing the number 5 as a sum of 1s and 2s in all possible ways in your notebook! How many ways did you find? (You should find 8 different ways!) Can you figure out the answer without listing down all the possibilities? Can you try it for n = 8?

Answer

There are exactly 8 ways for n = 5:

1 + 1 + 1 + 1 + 1	1 + 1 + 1 + 2	1 + 1 + 2 + 1	1 + 2 + 1 + 1
1 + 2 + 2	2 + 1 + 1 + 1	2 + 1 + 2	2 + 2 + 1

Without listing: put a “1 +” in front of every 4-beat rhythm (5 of them) and a “2 +” in front of every 3-beat rhythm (3 of them).

ways(5) = ways(4) + ways(3) = 5 + 3 = 8
ways(8) = ways(7) + ways(6) = 21 + 13 = 34

Why it happens: the first syllable is either short (1 beat) or long (2 beats). Those two cases never overlap and together they cover everything, so the counts simply add. This gives the Virahāṅka sequence 1, 2, 3, 5, 8, 13, 21, 34, …

Q4.

Use the systematic method to write down all 6-beat rhythms, i.e., write 6 as the sum of 1’s and 2’s in all possible ways. Did you get 13 ways?

Answer

Yes — exactly 13 ways. Write “1 +” before each of the eight 5-beat rhythms and “2 +” before each of the five 4-beat rhythms.

Starting with 1 + (8 ways)	Starting with 2 + (5 ways)
1 + 1 + 1 + 1 + 1 + 1	2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 2	2 + 1 + 1 + 2
1 + 1 + 1 + 2 + 1	2 + 1 + 2 + 1
1 + 1 + 2 + 1 + 1	2 + 2 + 1 + 1
1 + 1 + 2 + 2	2 + 2 + 2
1 + 2 + 1 + 1 + 1	—
1 + 2 + 1 + 2	—
1 + 2 + 2 + 1	—

8 + 5 = 13 ✓

Check it yourself: add up the beats in any row — every single sum comes to 6.

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