Page 19 - Ganit Kaushal

Application of Virahānka’s Sequence

The nth term of the Virahānka sequence provides the total number of distinct methods in which an
individual can climb ‘n’ steps, provided that they are permitted to ascend one step each time or a maximum
of two steps.
For Example: if you want to climb up 4 steps, then the total number of different ways in which you can do
it, is given by the 4th term of Virahānka sequence, provided you are allowed to climb one step upwards each
time or a maximum of two steps upwards. i.e., you can do it in 5 ways as the fourth term of the Virahānka
sequence is 5.
These five ways are shown below mathematically as well as pictorially:

4 = 1 + 1 + 1 + 1 (By taking one step upwards each time)
= 2 + 2 (By taking two steps upwards each time)

= 2 + 1 + 1 (First take two steps upwards, then one each time)
= 1 + 1 + 2 (First take one step then again one step and, then two steps upwards)

= 1 + 2 + 1 (First take one step upwards, then two steps and then one step again upwards)

Sequence of Hexagonal Numbers

A hexagonal number is a type of polygonal number that represents a pattern of dots arranged in the shape
of a hexagon.
These numbers can be visualised by forming a hexagon with dots and continuing to expand it layer by layer.

(Hexagonal number pattern)

1 7 19 37
The first four terms of the sequence are: 1, 7, 19, 37, …… (Hexagonal numbers)

Can you observe the pattern here?
Wow, you got it right. It starts with 1 and a continuously increasing multiple of six is added to the
previous term each time to get the next hexagonal number.
Observe that: 1 + 1 × 6 = 7 (First multiple of six is added to the first term, gives the 2nd term)

7 + 2 × 6 = 19 (Second multiple of six is added to the second term, gives the 3rd term)
19 + 3 × 6 = 37 (Third multiple of six is added to the third term, gives the 4th term),

and so on.

Patterns in Mathematics 17