Page 18 - Ganit Kaushal

In this picture, we have shown the first four terms of the pattern. In the fourth term, a total of 27 dots ̶ 9 black,
9 red and 9 blue are shown, as 9 added to itself 3 times gives 27. Readers are advised to make such a picture
for other terms of the pattern.

The powers of 3 sequence can also be given by the following formula for its n-th term T as:
n
T = 3 n – 1 ; n = 1, 2, 3, 4 ……
n
Sequence of Virahānka Numbers (Fibonacci Sequence)

Acharya Virahānka was an Indian prosodist (NUn'kkLrzh), who is widely known for his work in mathematics. He
was the first to suggest the numerical sequence that provided the rule for determining the number of variants
of ‘matra-vrattas’ in Sanskrit shlokas. The sequence, known as ‘The Sequence of Virahānka Numbers’, is
named after him.

Although Virahanka pre-dates Fibonacci, an Italian
mathematician, by several centuries, the sequence
gained general recognition around the world only
after Fibonacci’s 1202 work, ‘Liber Abaci’ (book 13
of calculation), in which he used the same pattern
to calculate rabbit population growth. Thus, 21
Virahanka and Fibonacci numbers are identical, yet
they represent distinct cultural origins of the same 2
mathematical concept, demonstrating mathematics’ 3 1
universal nature. 8
5
The Fibonacci (or Virahānka) sequence is:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ……

Can you observe the pattern?
Yes! The pattern followed here is that its first two terms are fixed as 1 and 2, thereafter each term is
obtained by adding the previous two terms.
The Fibonacci (Virahānka) number sequence can also be explained by the following recursive formula for its
n-th term:
T = 1, T = 2 and T = T n – 1 + T n – 2 ; n = 3, 4, 5, ….
n
2
1
Where T is the n-th term; T n – 1 is the preceding term and T n – 2 is the term before the preceding one.
n
Explanation of the Formula

For the 3rd term, put n = 3 in the recursion formula;
T = T + T = 2 + 1 = 3;
3
2
1
For the 4th term, put n = 4 in the recursion formula; Note
T = T + T = 3 + 2 = 5; For the Fibonacci/Virahanka number
3
2
4
sequence, some authors take T = 1; T = 1;
2
1
For the 5th term, put n = 5 in the recursion formula; then the sequence becomes 1, 1, 2, 3, 5, 8, 13,
T = T + T = 5 + 3 = 8; and so on.
5
4
3
For the 6th term, put n = 6 in the recursion formula;
T = T + T = 8 + 5 = 13; and so on.
6
5
4
16 Mathematics-6