The complete pattern in this 6-stellar is at follows:
0+1= 1
1+12=13
13+24=37
37+36=73
Therefore,
S5 = 73+ 48=121
S6 = 121 + 60= 181
Similarly to the triangular numbers, the 6-stellar sequences uses the same method but this time the number of dots can be found by preceding term added to by the multiples of 12 (as shown in red). Other way to write the pattern is in this way:
(1+0(12)), (1+1(12)), (1+3(12)) , (1+6(12)) , …
As soon as I wrote it this way I realized that there is clear relationship between the Triangular numbers and the way the 6-stellar numbers term.
Task 3
Find an expression for the 6-stellar number at S7
Now that we know the pattern for the first stages it is easy to recognize the stage S7
S7 = S6 + 12-1) (where S6 =181 and =7)
= 181+12(7-1)
= 181+12(6) (multiple of 12)
= 181+72
= 253
Task 4
Find the general statement for the 6-stellar number at stage Sn in terms of
To find the general statement we must look back to the Triangular number equation of , at this point we need to modify the equation so that it is suitable for the 6-stellar. So, in this case =12n because of the multiples of 12, the ( to because each of the numbers multiplied by 12 is one less than the of the term, and add 1 because the addition of terms starts initially from 1. Following all the rules the general statement should look like this:
Task 5
Repeat the steps above for other values of
A) Here is the number of dots for the sequence of 5-stellar number
S1 = 0+1 = 1
S2 = 1+10= 11
S3 = 11+20= 31
S4 = 31+30= 61
Therefore,
S5 = 61+40= 101
S6 = 101+50= 151
Similarly to the triangular numbers, the 5-stellar sequences uses the same method but this time the number of dots can be found by preceding term added to by the multiples of 10 (as shown in red). Other way to write the pattern is in this way:
(1+0(10)), (1+1(10)), (1+3(10)) , (1+6(10)) , …
As soon as I wrote it this way I realized that there is clear relationship between the Triangular numbers and the way the 5-stellar numbers term.
Now that we know the pattern for the first stages it is easy to recognize the stage S7
S7 = S6 + 10-1) (where S6 =151 and =7)
= 151+10(7-1)
= 151+10(6) (multiple of 10)
= 151+ 60
= 211
As done with the 6-stellar number, to find the general statement we must look back again to the Triangular number equation of , at this point we need to modify the equation so that it is suitable for the 5-stellar. So, in this case =10n because of the multiples of 10, the ( to because each of the numbers multiplied by 10 is one less than the of the term, and add 1 because the addition of terms starts initially from 1. Following all the rules the general statement should look like this:
B) Here is the number of dots for the sequence of 8-stellar number
S1 = 0+1 = 1
S2 = 1+16= 17
S3 = 17+32= 49
S4 = 49+48= 97
Therefore,
S5 = 97+64= 161
S6 = 161+80= 241
Similarly to the triangular numbers, the 8-stellar sequences uses the same method but this time the number of dots can be found by preceding term added to by the multiples of 16 (as shown in red). Other way to write the pattern is in this way:
(1+0(16)), (1+1(16)), (1+3(16)) , (1+6(16)) , …
As soon as I wrote it this way I realized that there is clear relationship between the Triangular numbers and the way the 8-stellar numbers term.
Now that we know the pattern for the first stages it is easy to recognize the stage S7
S7 = S6 + 16-1) (where S6 =241 and =7)
= 241+16(7-1)
= 241+16(6) (multiple of 16)
= 241+ 96
= 337
As done with the 6 and 5-stellar numbers, to find the general statement we must look back again to the Triangular number equation of , at this point we need to modify the equation so that it is suitable for the 8-stellar. So, in this case =16n because of the multiples of 16, the ( to because each of the numbers multiplied by 16 is one less than the of the term, and add 1 because the addition of terms starts initially from 1. Following all the rules the general statement should look like this:
Task 7
Produce a general statement; in terms of p and n, that generates the sequence of p-stellar numbers for any of the values of p at stage Sn
Comparing the results of the 5, 6 and 8 stellar numbers I could realize that the pattern of these numbers are identical except for the first number, which varies among sequences. We can also see that all the first term is usually equal to their p values. Hence, the general statement for p and n is:
Sn = 1+ (
Task 8
Test the validity of the general statement
Since we now have a general statement we can ensure that it works on larger numbers
For example: if we have and we want to know S8, then . Using our general statement we can get a quicker and easier answer:
S8 =
S8 = 1+ 840(28)
S8 = 1+ 23520
S8 = 23521 (At S8 the 420-stellar will have this amount of points)
Task 9
Explain how you arrived at the general statement
In order to arrive at the general statement I did first take the general statement of the triangular numbers on task 1 and then adjusted as a arithmetical sequence so that task 2 could be done. From the general statement of the 6-stellar number, I could now adjust the equation in order to get different values of p and its sequences. After doing the examples of 6,7 and 8 stellar numbers, I could therefore, understand the behaviour of the sequences by considering the relation of p and n. Following this relation, I was able to put either p or n in the general statement to by then take any data, no matter if it is a small or big value, as well as the geometrical shape of the number.
