Stellar Numbers (Part 1)
We are given the following diagrams which show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers. (1,3,6,)…
Looking at this triangular numbers we can link them to sequences where each triangle is sum of numbers. Next thing we can see is that every next triangle have one more row which is added to the previous triangle. This information which can be seen from these triangles can be put in table.
By using this table it is easier to work with data. So if each number of dots is sum of certain part of sequence we get that:
S1=1; S2=3, S3=6, S4=10, S5=15
Since each number represent sum, we should find actual numbers in this sequence and from general formula for sum which says that: Sn=U1+U2 + U3 + U4 + … + Un we can calculate these five numbers in this sequence: 1, 3, 6, 10, 15 and put them in table like it was shown above.
So; if Sn=U1+U2 + U3 + U4 + … + Un then S2=U2+U1 and since we have S2 which is 3 and we know U1 which is 1 we can calculate U2; U2= S2 -U1, we can therefore say that: U2=3-1 so we can calculate this and U2=2.
So; if Sn=U1+U2 + U3+U4+…+Un then S3=U1+U2+U3 and since U1+U2=S2 then S3=S2+U3, so U3=S3-S2 and from this U3=6-3 we can calculate that U3=3.
So; if Sn=U1+U2 + U3+U4+…+Un then S4=U1+U2+U3+U4 and since U1+U2+U3 =S3 then S4=S3+U4, so U4=S4-S3 and from this U4=10-6 we can calculate that U4=4.
So; if Sn=U1+U2 + U3+U4+…+Un then S5=U1+U2+U3+U4+U5 and since U1+U2+U3+U4=S4 then S5=S4+U5, so U5=S5-S4 and from this U5=15-10 we can calculate that U5=5.
From this information we can calculate difference in this sequence which can be determined by general formula which says d=Un-Un-1 and this formula is good only if sequence is arithmetic and difference is same between each number in sequence, this can be checked only by calculating three or more differences.
So, if we want to calculate difference between first and second number we take info from our earlier calculation and we get that U2=2 and U1=1 and then calculate d1=2-1 and difference between first and second number is d1=1.
If we want to prove that sequence is arithmetic we must calculate difference between second and third number so we take info from our earlier calculation and we get that U3=3 and U2=2 and then calculate d2=3-2 and difference between first and second number is d2=1. Since d2 and d1 are the same we can assume that this sequence is arithmetic but to be sure I will calculate d3 and d4.
Since d=Un-Un-1 then d3=4-3 => d3=1 and d4=5-4 => d4=1.
With these calculations we proved that this sequence is arithmetic and that the difference between each of them is 1. With this information we know that each new triangle has new row which is larger by one dot each time. And since this is typical arithmetic sequence we have general formula for sum of sequence terms which says that Sn= n/2 * (U1+Un) we can calculate how much dots have any triangle in this sequence.
For example if we want to calculate how many dots have the 15th triangle in this sequence we will calculate it like this:
First of all we will find what is 15th term in sequence and we will do that like this: Un=U1+(n-1)d so for the 15th term in sequence is U15=1+(15-1)·1 => U15=1+14 => U15=15 and then we use formula for Sn and calculate number of dots in 15th triangle. So, calculation of Sn is like this: S15= 15/2 · (1+15) => Sn= 15/2 · 14 => Sn=15 · 7 => S15=105.
So with this we can easily draw the three more triangles and calculate how many dots they have:
So for the 6th triangle we will calculate number of dots like this: Since Un=U1+(n-1)d we can calculate number of dots in new row: U6=1+(6-1)1 => U6=1+5 => U6=6 and with this info we can calculate the number of dots in whole triangle. Hence Sn= n/2 ·(U1+Un) then S6= 6/2 · (1+6) => S6= 3 · 7 => S6= 21.
For the 7th triangle we will calculate number of dots like this: Since Un=U1+(n-1)d we can calculate number of dots in new row: U7=1+(7-1)1 => U7=1+6 => U6=7 and with this info we can calculate the number of dots in whole triangle. Hence Sn= n/2 · (U1+Un) then S7= ·(1+7) => S7= 7/2 · 8 => S7= 7 · 4 => S7= 28.
For the 8th triangle we will calculate number of dots like this: Since Un=U1+(n-1)d we can calculate number of dots in new row: U8=1+(8-1)1 => U8=1+7 => U8=8 and with this info we can calculate the number of dots in whole triangle. Hence Sn= n/2 · (U1+Un) then S8= 8/2 · (1+8) => S8= 4 · 9 => S8= 36.
Since we calculated number of dots in new row and the number of dots in whole triangle for next three triangles we can draw them: