1+3+5+7+....+(
The following formula can be used to find the sum of the series:
Sn=
Sn =
Sn =
Sn =
Sn= n2 Where n
Therefore the relationship between ‘n’ and the number of dots is represented by the general statement: n2
Validation
Term number 4 → 42 = 16 =number of dots
Term number 2 →22= 4 = number of dots
Stellar numbers:
Working for the 6 stellar number:
Where p=6
Figure 3:
Table 3:
Expression for 3-stellar number at stage S4 = 1+ 12 + 24 + 36
= 73
Expression for 5-stellar number at stage S6 = 1 + 12 + 24 + 36 + 48 + 60
=181
Expression for 6-stellar number at stage S7 = 1 + 12 + 24 + 36 + 48 + 60 + 72
=253
After an analysis of the relationship between ‘n’ and the number of dots in each stage of the sequence, the number of dots in each stage could be expressed differently, as follows:
Table 4:
From the new expressions, I noticed that each numerical addition was a multiple of 12. Therefore the new expression could be modified slightly in order to discern a final pattern. The expression was modified in each case as follows.
Table 5:
The final expression followed a distinct pattern in each stage of stellar numbers. Each expression had an anomaly that remained the same in each stage, the numerical 1. Furthermore, the rest of the expression was a multiple of six that was multiplied by a number that was one greater than in the previous stage. In other words, the highlighted portion is the sum of numbers in an arithmetic progression. So they could be expressed as in Table 6.
Table 6:
The pattern was easily identified after this step and the general statement was identified. The anomaly numerical 1 remained the same in all the stages while the second numerical was ‘n’ itself. The numbers inside the second set of brackets were 6 in all the stages multiplied by one less than ‘n’. The pattern that repeated in all the stages could be generalized as:
1+ Where n
i.e.
This general statement was validated by substituting ‘n’ as any of the terms and ensuring that the final answer derived from using the general statement was equal to the number of dots in the stage.
Validation
Where ‘n’ is substituted as 3,
1+
=1+
=1+
=1+
=1+36
=37
When rechecking the value with the number of dots in figure 4 we see that they are the same.
Where ‘n’ is substituted as 6,
1+
=1+
=1+
=1
=1+180
=181
When corroborating the value with the number of dots in figure 5 we see that they are the same. Therefore we can say that the general statement derived for the number of dots in each stage of 6-stellar numbers is validated.
9-Stellar Numbers
Where p=9
Figure 6:
Table 7:
The first expression shows the calculation which leads to the total number of dots in each stage. In each stage, it is seen that the total dots increase by a multiple of 18. The primary expression can be expressed in a second way which involves an anomaly of 1 (therefore it must be present in the general statement for this sequence), multiples of 18 and ‘n’. The final expression of the number of dots shows an expression which can easily be generalized. The number 1 remains constant in the expression of each stage while the second number is the same as ‘n’ and the final number in the expression is the multiplication of 9 into one less than ‘n’.
The expression can be generalized to:
Where n
The following examples substitute ‘n’ in the general statement to test its validity:
Where ‘n’ is substituted as 1
=
=
=
= 1+0
= 1
Where ‘n’ is substituted as 4,
=
=
=
= 1+108
= 109
In both cases the substitution of ‘n’ in the general statement yielded the same result as the number of dots in the respective diagrams of stage n. Therefore the general statement is proved correct.
18-stellar numbers
Where p= 18
Figure 9:
Table 8:
After studying the relationship between the expression that represent the number of dots, we can see that each increase in the stage or term number causes as increase in the number of dots by a multiple of 36 which is essentially 2p or 2×18. The expression can be further modified to give us an expression that includes the anomaly 1, ‘n’ and multiples of 18. This leads us to the final expression that can be generalized to give us the statement:
Where n
We can verify the general statement by substituting ‘n’ with actual values.
Where ‘n’ is substituted by 3,
=
=
=
=1+108
=109
Where ‘n’ is substituted by 5,
=
=
=
=1+360
=361
In both of the above examples the general statement has yielded the correct result thereby validating the statement.
General statement of Stellar Numbers:
Table 9:
From observing the general statement of each of the stellar numbers, we can see that they are essentially the same. However we can also see that the general statements include the value of p. Stellar numbers correspond to the number of dots in a star diagram and they can be represented by pSn , where ‘n’ represents the stage of the stellar number and ‘p’ represents the number of vertices. So we can conclude that the general statement of p stellar numbers, pSn is:
The value of ‘n’ belongs to the set of natural numbers because ‘n’ is the first stage is represented by 1 dot and there can be no stage before this value, therefore n≠0. Due to the same reason, n cannot be negative or a decimal.
The variable ‘p’ may take any value except 0, 1, and 2 because the star with the least number of vertices is a three pointed star. Because ‘p’ represents the number of vertices of the stellar number, it cannot take negative values or decimal values either.
In order to validate the general statement we can substitute n and verify if the answer from the general statement matches the previously counted number of dots.
Where ‘p’ is substituted as 9 and ‘n’ is substituted as 4,
p=9, n=4
=
=
=
=1+108
=109
Where ‘p’ is substituted as 6 and ‘n’ is substituted as 3, 6S3
p=6, n=3
=
=
=
=1+36
=37
Where ‘p’ is substituted as 18 and ‘n’ is substituted as 5, 18S5
p=18, n=5
=
=
=
=1+360
=361
In all 3 examples, the general statement was found to be correct so the statement was verified.
Scope & Limitations:
The general formula may have various limitations although the general formula may have been proved correct for a range of numbers. Firstly, the general formula will be limited to numbers that are greater than 2, where p≠0, p≠1 and p≠2. This can be explained because ‘p’ represents the number of points the star possesses. There are no stars which can be formed with 0, 1 and 2 vertices. Therefore, these numbers cannot be used in the general statement.
Furthermore, ‘n’ cannot be represented by 0 because the first stage of the p-stellar number is 1 dot. There cannot be any stages preceding the first stage, represented by 1 dot.
Also, both ‘n’ and ‘p’ cannot represent negative numbers. This is simply because the number of vertices of any shape and the stage of the p-stellar number cannot be negative.
Figure 15 shows a 3 pointed star. This star has the least number of vertices. Therefore the general statement must keep in mind that ‘p’ belongs to the set of natural numbers and p>2 and that n>0.
General Statement:
Where p>2 and that n>0
Finally, if the number of dots in each stage change (eg. increase by 1), the general statement of the series changes as well. The number of dots used to compose the same figure (ie. the same value of ‘p’) can be changed which will affect the general statement.
This occurrence can be illustrated using the following figures:
Figure 16:
Table 10:
The series can also be written as, excluding term 1 because it does not follow the same pattern as the rest of the series:
24+36+48+60+...+12
To find the sum of the ‘n’ terms of this series, the following formula can be used:
Where = first term of the series and = nth term
Where n
However, because each ‘number of dots’ which corressponds to the term number, has a constant value of 1 (as seen in the table), 1 is added to the final general statement.
Where n
Validation
Where ‘n’ is substituted as 1
Where ‘n’ is substituted as 3
Applications of Stellar Numbers:
Stellar numbers like other sequences and series could be used in various everyday situations. For example, an investigation of series can be used to convert repeating decimals to fractions or to calculate the compound interest on a property. Stellar numbers could also be applied to architectural designs. For example, pipes could be shaped into star shapes in order to increase the surface area from which heat from hot water could be lost in order to heat the surroundings. The star shape would also decrease the cross sectional area compared to a circular pipe in order to change the pressure of the water flowing through the pipes. The same concept could be used in chemical reactors to increase the amount of heat supplied to a reaction.