An expression for the stellar number at S7 , the 7th term is S7= S6+ (S6-S5) + 12 (where 12 is the value of the second differences). This was derived through the pattern noticed above where essentially the sum of each row is equivalent to the next Sn value, or at any value of n, where the second differences exist (), adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
The general statement for the 6-stellar number at stage Sn in terms of n (where n is an element of positive integers) can be found in a variety of ways. If one were to want a recursive formula, from the pattern found earlier to get S7, it would be
Sn =2(Sn-1)-Sn-2+12.;+. However, this equation is only useful when n is greater than or equal to three. Say: Will be discussed later.
Another way of finding the general statement is to look at the differences again. Since the second differences are constant and not zero, the equation would be to the second degree, and therefore can be expressed with the general formula for any quadratic equation, being: Sn=. From here, Sn will be represented by r
To solve for a, one must isolate it in the equation, ag!=s, where s is the value of the constant differences, and g is the degree of the function. In this case, g=2, and s=12.
Now, by substituting in 2 ordered pairs in the equation, two equations are created with only b and c as the unsolved variables in each. From here there is a system of equations that must be solved, and then the values of b and c are found.
Substitute equation [1] into equation [2]:
Substitute c=1 into equation [1]
b=-5-(1)
b=-6
Therefore the general statement for the 6-stellar number at stage Sn is Sn , where n is an element of positive integers. The last way that this general formula could be found is with a graphing calculator. Since it has already been established the general statement is to the degree two, therefore, by using the quadratic regression function of the calculator, the a, b and c values will be given by the calculator.
Now the task asks that the above steps are repeated for other values of p, shown below. It must be noted that in order for a star to be formed, creating stellar numbers, the sequence must have a p value of at least 3, because at positive integer values of 1 and 2, a star is not formed. Therefore is an element or positive integers.
First p=5 is considered. Below are stages 1-6
S1 S2 S3 S4
S5 S6
Now here is a chart showing the number of dots (the stellar number) in these stages, when p=5. Because these stellar numbers can be considered a sequence (where n is the term number, or the stage and Sn is the number of dots or the stellar number, and n is an element of whole numbers) or a relation, it also shows up to the third differences.
When the number of dots was counted, it was noticed that for each added layer there were ten more dots in that layer. Then after organizing the data into the above table, the value of the second differences (the degree of the relation created by these stellar numbers) was also 10. 10 is two times the number of vertices in this case. Also it is noticed that once the second differences do exist (starting from when n= 3), at that n value, adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
An expression for the 7th term is S7= S6+ (S6-S5) + 10 (, where 10 is the constant second difference). Again, this was derived through the pattern noticed above where essentially the sum of each row is equivalent to the next Sn value, or at any value of n, where the second differences exist, and n is greater than or equal to three., adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
The general statement for the 5-stellar number at stage Sn in terms of n (where n is an element of positive integers) is found here in multiple ways. If one were to want a recursive formula, from the pattern found earlier to get S7, it would be in simplified form,
Sn =2(Sn-1)-Sn-2+10 , +
Another way of finding the general statement is to look at the differences again. Since the second differences are constant and not zero, the equation would be to the second degree, and therefore can be expressed with the general formula for any quadratic equation, being: Sn=. From here Sn will be represented by l
To solve for a, one must isolate it in the equation, ag!=s, where s is the value of the constant differences, and g is the degree of the function. In this case, g=2, and s=10.
Now, by substituting in 2 ordered pairs in the equation, two equations are created with only b and c as the unsolved variables in each. From here there is a system of equations that must be solved, and then the values of b and c are found.
Substitute equation [1] into equation [2]:
Substitute c=1 into equation [1]
b=-4-(1)
b=-5
Therefore the general statement for the 5-stellar number at stage Sn is Sn+. Again, the last way that this general formula could be found is with a graphing calculator. Since it has already been established the general statement is to the degree two, therefore, by using the quadratic regression function of the calculator, the a, b and c values will be given by the calculator.
Now p= 7 is considered. Below are stages 1 to stage 6.
Here is a chart showing the number of dots (the stellar number) up to stage 6, when p=7. Because these stellar numbers can be considered a sequence (where n is the term number, or the stage and Sn is the number of dots or the stellar number, and n is an element of whole numbers) or a relation, it also shows up to the third differences.
When the number of dots was manually counted, it was noticed that for each added layer there were ten more dots in that layer. Then after organizing the data into the above table, the value of the second differences (the degree of the relation created by these stellar numbers) was also 14. 14 is two times the number of vertices in this case. Again it is noticed that once the second differences do exist (starting from when n= 3), at that n value, adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
An expression for the 7th term is S7= S6+ (S6-S5) + 14 (where 14 is the value of the second differences). Again, this was derived through the pattern noticed above where essentially the sum of each row is equivalent to the next Sn value, or at any value of n, where the second differences exist, (when n is greater than or equal to three), adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
The general statement for the 7-stellar number at stage Sn in terms of n (where n is an element of positive integers) is found here in multiple ways. If one were to want a recursive formula, from the pattern found earlier to get S7, it would be, in simplified form,
Sn =2(Sn-1)-Sn-2+14,+
Another way of finding the general statement is to look at the differences again. Since the second differences are constant and not zero, the equation would be to the second degree, and therefore can be expressed with the general formula for any quadratic equation, being: Sn=. From here Sn will be represented by v
To solve for a, one must isolate it in the equation, ag!=s, where s is the value of the constant differences, and g is the degree of the function. In this case, g=2, and s=14.
Now, by substituting in 2 ordered pairs in the equation, two equations are created with only b and c as the unsolved variables in each. From here there is a system of equations that must be solved, and then the values of b and c are found.
Substitute equation [1] into equation [2]:
Substitute c=1 into equation [1]
b=-6-(1)
b=-7
Therefore the general statement for the 7-stellar number at stage Sn is Sn
Again, the last way that this general formula could be found is with a graphing calculator. Since it has already been established the general statement is to the degree two, therefore, by using the quadratic regression function of the calculator, the a, b and c values will be given by the calculator.
Next p=10 is considered. Below is the pattern from stages 1-6.
S1 S2 S3 S4
S5 S6
Below is a chart showing the number of dots (the stellar number) up to stage 6, when p=10. Because these stellar numbers can be considered a sequence (where n is the term number, or the stage and Sn is the number of dots or the stellar number, and n is an element of whole numbers) or a relation, it also shows up to the third differences.
While manually counting dots, it was noticed that for each added layer there were twenty more dots in that layer. Then after organizing the data into the above table, the value of the second differences (the degree of the relation created by these stellar numbers) was also 20. 20 is two times the number of vertices in this case. Also it is noticed that once the second differences do exist (starting at n =3), at that n value, adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage
When p is 7, an expression for the 7th term is S7= S6+ (S6-S5) + 20 ( 20 being the value of the second differences). Essentially the sum of each row is equivalent to the next Sn value, or at any value of n, where the second differences exist (), adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
The general statement for the 10-stellar number at stage Sn in terms of n (where n is an element of positive integers) is found here in three different ways. Firstly, if one were to want a recursive formula, then from the pattern found earlier to get S7, it would be, in simplified form,
Sn =2(Sn-1)-Sn-2+20,
Another way of finding the general statement is to look at the differences again. Since the second differences are constant and not zero, the equation would be to the second degree, and therefore can be expressed with the general formula for any quadratic equation, being: Sn=. From here Sn will be represented by u
To solve for a, one must isolate it in the equation, ag!=s, where s is the value of the constant differences, and g is the degree of the function. In this case, g=2, and s=20.
Now, by substituting in 2 ordered pairs in the equation, two equations are created with only b and c as the unsolved variables in each. From here there is a system of equations that must be solved, and then the values of b and c are found.
Substitute equation [1] into equation [2]:
Substitute c=1 into equation [1]
b=-9-(1)
b=-10
Therefore the general statement for the 6-stellar number at stage Sn is Sn+
Again, the last way that this general formula could be found is with a graphing calculator. Since it has already been established the general statement is to the degree two, therefore, by using the quadratic regression function of the calculator, the a, b and c values will be given by the calculator.
Lastly, p=13 is considered.
S1 S2 S3
S4 S5 S6
Below is a chart showing the number of dots (the stellar number) up to stage 6, when p=13. Because these stellar numbers can be considered a sequence (where n is the term number, or the stage and Sn is the number of dots or the stellar number, and n is an element of whole numbers) or a relation, it also shows up to the third differences.
While counting stars, it was noticed that for each added layer there were 26 more dots in that layer. Then after organizing the data into the above table, the value of the second differences (the degree of the relation created by these stellar numbers) was also 26. 26 is two times the number of vertices in this case. Also it is noticed that once the second differences do exist (starting from when n= 3), at that n value, adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
An expression for the 7th term is S7= S6+ (S6-S5) + 26). This was derived through the pattern noticed above where essentially the sum of each row is equivalent to the next Sn value, or at any value of n, where the second differences exist (), adding the Sn value at that value of n, the 1st differences at that value, and the 2nd differences at that value, the sum is equivalent to the value of Sn in the next stage.
The general statement for the 13-stellar number at stage Sn in terms of n (where n is an element of positive integers) is found here in multiple ways. If one were to want a recursive formula, from the pattern found earlier to get S7, it would be, in simplified form,
Sn =2(Sn-1)-Sn-2+26+.
Another way of finding the general statement is to look at the differences again. Since the second differences are constant and not zero, the equation would be to the second degree, and therefore can be expressed with the general formula for any quadratic equation, being: Sn=. From here Sn will be represented by o
To solve for a, one must isolate it in the equation, ag!=s, where s is the value of the constant differences, and g is the degree of the function. In this case, g=2, and s=26.
Now, by substituting in 2 ordered pairs in the equation, two equations are created with only b and c as the unsolved variables in each. From here there is a system of equations that must be solved, and then the values of b and c are found.
Substitute equation [1] into equation [2]:
Substitute c=1 into equation [1]
b=-12-(1)
b=-13
Therefore the general statement for the 13-stellar number at stage Sn is Sn+
Again, the last way that this general formula could be found is with a graphing calculator. Since it has already been established the general statement is to the degree two, therefore, by using the quadratic regression function of the calculator, the a, b and c values will be given by the calculator.
Next the task asks for a general statement in terms of p and n that will generate the sequence of p-stellar numbers for any value of p at stage Sn. The answer to this is Sn=, where p is an element of positive integers greater than or equal to 3, and n is an element of positive integers. A recursion formula could work as well, which would be Sn= 2Sn-1+Sn-2+2n, but this formula is somewhat inefficient in comparison to the other. It was noticed that in the general formula of any quadratic equation (Sn=), the a was always the p value, and the b value was always the negative of the p value. Thus this formula was derived.
Next the validity of the general statement was tested. Note: It is impossible to have equation editor pleasenumber of vertices, negative number of vertices, rational and irrational number of vertices, etc, so these values could not be tested in the general statement.
Let p=9, n= 5
When manually counted there were 181 dots
Substitute p=9, and n=5 into general statement:
S5=
Let p=11, n=9
When manually counted, there were 793 dots
Substitute these p and n values into general statement.
S9
Let p=18, n=11
The number was counted to be 1981. <My ass you counted that!
S11
Therefore, for all admissible values of p and n, the general statement is valid. p must be greater than or equal to three, where it is an element of positive integers and n must be an element of positive integers.
The limitations for the general statement are that for both n, and p, all negative integer values will not work because it is impossible to have half a term number, or -stellar numbers. Also it must be noted that n can not equal zero, simply because there can not be a zero term number. However, algebraically, when p is positive integer values one and two, the same patterns noticed above for other stellar numbers are noticed. Nevertheless, visually 2-stellar numbers do not work because a star is not formed. This is why integer values of 1 and 2 were deemed inadmissible. Below is a chart showing why algebraically this is valid, and why visually it is not. For p=2
(The 2nd differences are 2 times the p value; the recursion formula still applies, as well as other patterns are the same).
Yet, visually, only rhombii are formed, and therefore p=2 is inadmissible. Depends on how you define a stellar number. ☺
To arrive at the general statement, there had to have been many trials done to get the specific general statement for different p-values to notice that in the general equation for each of those values, the a and b values were the p value and the negative of the p value respectively. The c value was always one, no matter what the equation. For example, in Sn, this for for p=13, the c value is one, and it is evident that the a and b values are the p value and the negative of the p value respectively
Overall this task helps to further ones understanding of special numbers and their properties. From the ones considered in this task, it seems that the equations for all special numbers that create two-dimensional figures, the equations are to the second degree. More investigation should be done to prove this.
Header: Type 1, Stellar numbers, candidate #, mathematics SL > this is what IB survival told me
Footer: page ___ of ___ > again what IB survival told me
TITLE PAGE! (name candidate number type 1 stellar numbers school name) that should be enough
I recommend italicizing your variables and learn to use the equation editor. (Alt =). Also, be sure to format this better, state restrictions on your final equations (so post calculations e.g. if you’re saying THIS IS THE EQUATION FOR THIS). Also, for testing validity, use numbers in equations to test backed up with reasoning.
