NCERT Solutions Ganita Prakash (Part 1) Chapter 6 In-text Questions — The Virahāṅka–Fibonacci Numbers

Book page 142 Updated on2026-09-05

Q1.

Write the next number in the sequence, after 55.

Answer

The next number is 89.

Rule: each number = sum of the two numbers before it.
1 + 2 = 3 • 2 + 3 = 5 • 3 + 5 = 8 • 5 + 8 = 13
8 + 13 = 21 • 13 + 21 = 34 • 21 + 34 = 55
34 + 55 = 89

Check it yourself: the rule holds for every number already printed, which is why we can trust it for the next one.

Q2.

Write the next 3 numbers in the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ____, ____, ____, … If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Answer

The next three numbers are 144, 233, 377.

55 + 89 = 144
89 + 144 = 233
144 + 233 = 377

The number after 377 will be even, and you can tell without adding.

Parities so far: 1(O), 2(E), 3(O), 5(O), 8(E), 13(O), 21(O), 34(E), 55(O), 89(O), 144(E), 233(O), 377(O)
Last two are odd and odd → odd + odd = even

Why it happens: the parities repeat in the block odd, odd, even over and over. After two odds the next term must be even. (The actual value is 233 + 377 = 610, which is indeed even.)

Q3.

What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?

Answer

Position	1	2	3	4	5	6	7	8	9	10	11	12
Number	1	2	3	5	8	13	21	34	55	89	144	233
Parity	O	E	O	O	E	O	O	E	O	O	E	O

The pattern is O, E, O, O, E, O, O, E, … — after the first term the block odd, odd, even repeats forever.

O + E = O
E + O = O
O + O = E … and the cycle starts again

So a term is even exactly when its position is 2nd, 5th, 8th, 11th, 14th, … — that is, when the position leaves remainder 2 on division by 3.

Why it happens: the parity of a term depends only on the parities of the two before it. Since there are only a few parity pairs, the pattern must repeat — and here it repeats every three terms.

Q4.

How many petals do you see on each of these flowers?

Answer

Counting the petals in the three daisy pictures gives 13, 21 and 34.

Flower	Petals counted	Is it a Virahāṅka number?
First daisy	13	Yes — 6th term
Second daisy	21	Yes — 7th term
Third daisy	34	Yes — 8th term

And 13, 21, 34 are three numbers standing next to each other in the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Why it happens: petals, sunflower spirals and pine-cone spirals grow in a way that packs seeds and petals most efficiently, and this packing throws up Virahāṅka–Fibonacci numbers again and again in nature.

Did you know? next time you see a marigold, a sunflower or a pineapple, count the spirals — you will very often land on 8, 13, 21 or 34.

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