Maths Portfolio (Stellar Numbers)

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In order to develop this mathematics SL portfolio, I will require the use of windows paint 2010 and the graphic calculator fx-9860G SD emulator, meaning that I will use screenshots from this software with the intention of demonstrating my work and process of stellar numbers sequences. Triangular numbers are those which follow a triangular pattern, these numbers can be represented in a triangular grid of evenly spaced dots.

The sequence of triangular numbers is shown in the diagrams above. The first stage has 1 dot; the second stage has 3 dots (1+2); the third stage has 6 dots (1+2+3); the fourth stage has 10 dots (1+2+3+4); the fifth stage has 15 dots(1+2+3+4+5); the sixth stage has 21 dots (1+2+3+4+5+6) ; the seventh stage has 28 dots(1+2+3+4+5+6+7) and the eighth stage has36 dots(1+2+3+4+5+6+7+8). As it could be noticed, there is a sequence where in every stage the number of dots is obtained by adding up all the positive integers that correspond to the previous stages and every time one more number is added.

In terms of n, where n matches up to the stage number, it is accurate to establish an equation so that when trying to find the number of dots in stage 592, it is easy and fast by simply applying the following formulae:

Now it is possible to find the nth number by using the formula, going back to the example where n is 592, so we replace n by 592 and solve the equation as follows: so the 592nd term will contain 175528 dots. =175528

Furthermore, to prove my equation I will use different values for n but they have to be positive integers otherwise if I use negative or irrational or fractions it would not have any common sense. thus i will replace n by 6 to prove that the result is 21 as shown in the first diagram and also by 10:

=21 =55

Subsequently considering stellar numbers which are stars with 6 vertices, the number of dots in each stage will represent each stellar number just as it was done before with the...
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