In the above pattern of polygons, in the first shape (x = 1), we get a closed shape called a triangle. Similarly the second shape (x = 2 ) gives a four sided polygon and the third shape (x = 3) is a five sided polygon and continuing in the same way two more shapes are formed. If the number of match sticks required to form each of the shapes is taken as y, then the values of x and y are tabulated as given below.

Observe the table and express y in terms of x as below:
when x = 1, y = 3 = 1 + 2
when x = 2 , y = 4 = 2 + 2
when x = 3, y = 5 = 3 + 2
when x = 4 , y = 6 = 4 + 2
when x = 5, y = 7 = 5 + 2
Hence, each of the values of y which we get from the table is 2 more than x. That is
y = x + 2 .
Therefore,
6^th shape (x = 6) will have y = 8 = 6 + 2 (8 match sticks)
7^th shape (x = 7) will have y = 9 = 7 + 2 (9 match sticks)
8^th shape (x = 8) will have y = 10 = 8 + 2 (10 match sticks)
We clearly see that the next three shapes will require 8, 9 and 10 matchsticks.

The row sum of elements of a Pascal’s Triangle are shown below:

First row = 21 − 1 = 1
Second row = 22 − 1 = 2 x 1 = 2
Third row = 23 − 1 = 2 x 2 = 4
Fourth row = 24 − 1 = 2 x 2 x 2 = 8
Fifth row = 25 − 1 = 2 × 2 × 2 × 2 = 16
Sixth row = 26 − 1 = 2 × 2 × 2 × 2 × 2 = 32
Seventh row = 27 − 1 = 2 × 2 × 2 × 2 × 2 × 2 = 64
Eighth row = 28 − 1 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
Here x denotes the row and y denotes the corresponding row sum. The values of x and y can be tabulated as follows:

The relationship between x and y is y = 2^{x – 1.}

Observe the table carefully. To verify the relationship between x and y, let us substitute the values of x and get the values of y.
If x = 1, y = \(\frac { 1(1+1) }{ 2 } =\frac { 2 }{ 2 } =1\)
If x = 2 , y = \(\frac { 2(2+1) }{ 2 } =\frac { 6 }{ 2 } =3\)
If x = 3, y = \(\frac { 3(3+1) }{ 2 } =\frac { 12 }{ 2 } =6\) 
If x = 4 , y = \(\frac { 4(4+1) }{ 2 } =\frac { 20 }{ 2 } =10\)
If x = 5, y =  \(\frac { 5(5+1) }{ 2 } =\frac { 30 }{ 2 } =15\)
Hence, y = \(\frac { x(x+1) }{ 2 } \) is verified.

8 × 45 × 84 = 28 × 9 × 120

1 × 13 × 66 = 11 × 1 × 78