Page 17 - Ganit Kaushal

It can be observed that the cube of side 2 units is made up of 8 small cubes of side one unit. So, the second
cubic number is 8 (yellow cubes in the given figure).

A cube of side 3 units is made up of 27 cubes of side 1 unit. So, the third cubic number is 27 (green cubes in
the given figure).
A cube of side 4 units is made up of 64 cubes of side 1 unit. So, the fourth cubic number is 64 (blue cubes in
the given figure). And we can move on and on.

The sequence of cubes can also be given by the following formula for its n-th term T as:
n
3
T = n ; n = 1, 2, 3, 4 ……
n
Sequence of Powers of 2

Consider the sequence 1, 2, 4, 8, 16 …… (Powers of 2)

Can you observe the pattern here?
Well, the pattern followed here is that it starts with 1, and each term is added to itself to obtain the next
term. Yes, you got it right, the next number in this pattern is 32 because 16 will be added to itself.
Is there any pictorial way to visualise the above sequence of powers of two?

Yes, we can give you one possible way of visualising the sequence of powers of 2. The above sequence
can be explained with the help of the following picture. The picture shows the first five terms of the
square pattern.

(Powers of 2 pattern)

1 2 4 8 16
The powers of 2 sequence can also be given by the following formula for its n-th term T as:
n
T = 2 (n – 1) ; n = 1, 2, 3, 4 ……
n
Sequence of Powers of 3

The sequence 1, 3, 9, 27, 81 …… (Powers of 3)

Can you observe the pattern here?
Well, the pattern followed here is that it starts with 1, and each term is added to itself three times to
obtain the next term. Observe, 1 added to itself three times (1 + 1 + 1 = 3) gives 3. Then, 3 added to
itself 3 times gives 9, then 9 added to itself 3 times gives 27, and so on.
Is there any pictorial way to visualise the above sequence of powers of 3?
Yes, the given picture gives you one possible way of visualising the powers of 3.

(Powers of 3 pattern)

1 3 9 27

Patterns in Mathematics 15