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# 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:

*S _{1}=1; S_{2}=3, S_{3}=6, S_{4}=10, *

*S*

_{5}=15Since each number represent sum, we should find actual numbers in this sequence and from general formula for sum which says that: * S_{n}=U_{1}+U_{2} + U_{3 }+ U_{4 }+ … + U_{n }*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 ** S_{n}=U_{1}+U_{2} + U_{3 }+ U_{4 }+ … + U_{n }**then

*and since we have*

**S**_{2}=U_{2}+U_{1 }**S**which is 3 and we know

_{2}**U**which is 1 we can calculate

_{1 }**, we can therefore say that:**

*U*_{2}; U_{2}= S_{2}-U_{1}**so we can calculate this and**

*U*_{2}=3-1

*U*_{2}=2.So; if *S_{n}=U_{1}+U_{2} + U_{3}+U_{4}+…+U_{n }*then

**and since**

*S*_{3}=U_{1}+U_{2}+U_{3 }**then**

*U*_{1}+U_{2}=S_{2 }*so*

**S**,_{3}=S_{2}+U_{3}**and from this**

*U*_{3}=S_{3}-S_{2 }**we can calculate that**

*U*_{3}=6-3

*U*_{3}=3*.*

So; if ** S_{n}=U_{1}+U_{2} + U_{3}+U_{4}+…+U_{n }**then

**and since**

*S*_{4}=U_{1}+U_{2}+U_{3}+U_{4 }**then**

*U*_{1}+U_{2}+U_{3}=S_{3 }**and from this**

*S*so_{4}=S_{3}+U_{4},*U*_{4}=S_{4}-S_{3 }**we can calculate that**

*U*_{4}=10-6

*U*_{4}=4.So; if * S_{n}=U_{1}+U_{2} + U_{3}+U_{4}+…+U_{n }*then

**and since**

*S*_{5}=U_{1}+U_{2}+U_{3}+U_{4}+U_{5 }**then**

*U*_{1}+U_{2}+U_{3}+U_{4}=S_{4 }

*S*so_{5}=S_{4}+U_{5},*and from this*

**U**_{5}=S_{5}-S_{4 }**we can calculate that**

*U*_{5}=15-10

*U*_{5}=5.From this information we can calculate difference in this sequence which can be determined by general formula which says ** d=U_{n}-U_{n-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 * U_{2}=2 *and

**and then calculate**

*U*_{1}=1**and difference between first and second number is**

*d*_{1}=2-1

*d*_{1}=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 ** U_{3}=3 **and

**and then calculate**

*U*_{2}=2*and difference between first and second number is*

**d**_{2}=3-2

*d*_{2}=1*.*Since

**d**and

_{2}**d**are the same we can assume that this sequence is arithmetic but to be sure I will calculate

_{1}**d**and

_{3}**d**

_{4.}Since ** d=U_{n}-U_{n-1 }**then

**d**and_{3}=4-3 => d_{3}=1**d**_{4}=5-4 => d_{4}=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** S _{n}= n/2 * (U_{1}+U_{n}) **we can calculate how much dots have any triangle in this sequence.

For example if we want to calculate how many dots have the 15^{th} triangle in this sequence we will calculate it like this:

First of all we will find what is 15^{th} term in sequence and we will do that like this: * U_{n}=U_{1}+(n-1)d *so for the 15

^{th}term in sequence is

**and then we use formula for**

*U*_{15}=1+(15-1)·1 => U_{15}=1+14 => U_{15}=15**and calculate number of dots in 15**

*S*_{n }^{th}triangle. So, calculation of

**is like this:**

*S*_{n}

*S*_{15}= 15/2*·**(1+15) =>**S*_{n}= 15/2*·**14 =>**S*_{n}=15 · 7 => S_{15}=*105.*So with this we can easily draw the three more triangles and calculate how many dots they have:

So for the 6^{th} triangle we will calculate number of dots like this: Since ** U_{n}=U_{1}+(n-1)d **we can calculate number of dots in new row:

**and with this info we can calculate the number of dots in whole triangle. Hence**

*U*_{6}=1+(6-1)1 => U_{6}=1+5 => U_{6}=6**then**

*S*_{n}= n/2*·**(U*_{1}+U_{n})

*S*_{6}= 6/2*·**(1+6) =>**S*_{6}= 3*· 7 =>**S*_{6}= 21.For the 7^{th} triangle we will calculate number of dots like this: Since ** U_{n}=U_{1}+(n-1)d **we can calculate number of dots in new row:

**and with this info we can calculate the number of dots in whole triangle. Hence**

*U*_{7}=1+(7-1)1 => U_{7}=1+6 => U_{6}=7**then**

*S*_{n}= n/2*·**(U*_{1}+U_{n})

*S*_{7}=*·**(1+7) =>**S*_{7}= 7/2*· 8 =>**S*_{7}= 7 · 4*=>**S*_{7}= 28.For the 8^{th} triangle we will calculate number of dots like this: Since ** U_{n}=U_{1}+(n-1)d **we can calculate number of dots in new row:

**and with this info we can calculate the number of dots in whole triangle. Hence**

*U*_{8}=1+(8-1)1 => U_{8}=1+7 => U_{8}=8**then**

*S*_{n}= n/2*·**(U*_{1}+U_{n})

*S*_{8}= 8/2*·**(1+8) =>**S*_{8}= 4*· 9 =>**S*_{8}= 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:

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