Substitute the value of ‘a’ of one equation:

To find the General statement that represents the nth triangular number in terms of ‘n’, we substitute the value of ‘y’ by Un and the value of ‘x’ by n, thus:

## STELLAR NUMBERS

##
NUMBER OF DOTS TO S6 STAGE

Thus, using finite difference:

The most obvious pattern is that the 1st row all numbers and odd and the second row all are even.

Also all these numbers are some multiples of 12 + 1, for example: 12 also turns out to be the half of 6.

####
6 STELLAR NUMBER AT STAGE S7

1 + 1 (12) + 2 (12) + 3 (12) + 4(12) + 5(12) + 6(12) = 253

###
GENERAL STATEMENT FOR 6 STELLAR NUMBER AT STAGE SN IN TERMS OF N

If you notice the multiples are triangular numbers thus the general statement will, in some way contain the general statement of the triangle numbers. Because the final difference of the stellar numbers is 12 the multiple is Thus:

## 4 STELLAR NUMBER

## NUMBER OF DOTS AT 4 STELLAR

###
4 STELLAR NUMBER AT STAGE S7

1 + 1 (8) + 2 (8) + 3 (8) + 4(8) + 5(8) + 6(8) = 169 (8 being multiple of 2 and 4)

###
GENERAL STATEMENT FOR 4 STELLAR NUMBER AT STAGE SN IN TERMS OF N

Thus the pattern for 4 stellar, some multiples of 8 with triangular numbers:

Thus:

## 8 STELLAR NUMBER

## NUMBER OF DOTS AT 8 STELLAR

###
8 STELLAR NUMBER AT STAGE S7

1 + 1 (16) + 2 (16) + 3 (16) + 4(16) + 5(16) + 6(16) = 337 (Taking the double of 8)

###
GENERAL STATEMENT FOR 8 STELLAR NUMBER AT STAGE SN IN TERMS OF N

Thus the pattern for 8 stellar, some multiples of 16 with triangular numbers:

Thus:

## GENERAL STATEMENT IN TERMS OF P AND N

The general statement is:

### VALIDITY

1. Find the number of dots, in a star with 10 vertices when n = 5 2. Find the number of dots, in a star with 1 vertices when n = 5

3. Find the number of dots, in a star with 3 vertices when n = 3.5

#### PROFF

- 1 + 1 (20) + 2 (20) + 3 (20) + 4(20) = 201
- 1 = 1
- –

## LIMITATIONS OF THE GENERAL STATEMENT AND ARRIVAL AT GENERAL STATEMENT

Due to the fact that each vertex on the star has a triangular pattern on it, it is easy to determine the formula: you multiply the general statement of the triangular pattern by the number of vertices that the star has and then add 1. The reason for this addition is so that you can add the dot in the middle of the star.

The next difficulty faced is: if I multiply by the number of vertices the points on the closest will also be multiplied twice, and this could not be.

But then how do we add the inside star to the final result. In the example below the star on the inside has 12 dots (excluding the point in middle), and the outside star if you count the dots on the inside twice you will count a total of 12 dots, thus covering the dots of the inside star –

This is the same for all other stellar shapes.