# IB math portfolio - Triangular and Stellar numbers

by kamilaakshulakova (student)

Triangular numbers

The following  diagram shows a trinagular patterns of evenly spaced dots.The  number of dots in each diagram are examples of Triangular numbers :

In order to show each diagram I bring the name in terms of (T1,T2,T3..)  with number of dots in each first row of diagram. I notice that there is always one more row than in previous diagram , that row has one more dot than the previous one.With knowing that the difference between each diagram is the same, I assume that the diagrams are parts of the seguence.

T1                  T2                 T3                        T4                                 T5

In that form, I can determine that next diagrams can be completed by adding  an extra row with the same number of dots as the term number to the prevous one . For example, T7 will have the same amount of dots as T6 plus Term number(15 +6=21).Therefore, to the next three diagrams I will give name in order with amount of dots in extra row : T6, T7, T8.

T7                                         T8                                                T9

Now I will try to find a general formula using my knowlegde and observations about triangular numbers that will describe the progression of number of the dots:

I made this table to show that numbers of dots in each term is equal to the number of dots of that term plus the number of the previous term:

T1:1+2=3

T2:1+2+3 =6

T3:1+2+3+4=10

T4 :1+2+3+4+5=15.

I think it is clear example of arithmetic progression, and  to find a general statement I can use the formula from the way we find a sum of certain numbers of an arithmetic sequence:

Tn=n(a1+an)/2  where   an =a1 +(n-1)d, so I have

Tn= n(2a+d(n-1))/2 where a represents the number of dots in the first term, and d is ...