# Math Portfolio Type 1. In this portfolio I will be investigating geometric shapes that lead to certain special numbers.

In this portfolio I will be investigating geometric shapes that lead to certain special numbers. To help me gain insight into these shapes and related numbers I will be using Microsoft Word 2007, Microsoft Equation Editor 3.0, Microsoft Paint and Inkscape, which is a graphing software. Also to arrive at the mathematical conclusions, I will be applying the concept of sequences, arithmetic sequences in specific. The IB curriculum has taught me how to recognize patterns in sequences and series, which will help me solve and understand this task.

Considering this triangular pattern of evenly spaced dots, one can see that there are 1, 3, 6, 10, 15 dots in the successive stages, thus giving us a sequence to be studied further. But as observed, the terms of this sequence neither have a common ration nor a common difference, hence they do not fall under arithmetic or geometric sequences and because of this the formulas for an arithmetic or a geometric sequence cannot be applied at this point.

But if the pattern is observed carefully, it can be seen that the terms in this sequence are the sum of the first n positive integers.

Let u1, u2… un represent terms in the sequence.

u1 = 1

u2 = 3 = 1+2

u3 = 6 = 1+2+3

u4 = 10 = 1+2+3+4

u5 = 15 = 1+2+3+4+5

By looking at the working above, we can tell that un is the sum of the first n positive integers. Hence, by using the above logic we can find out the next three terms in this sequence.

u6 = 1+2+3+4+5+6 = 21

u7 = 1+2+3+4+5+6+7 = 28

u8 = 1+2+3+4+5+6+7+8 = 36

S6                                   S7                                               S8

As previously proved this sequence is the sum of the first n positive integers.

i.e.  un = 1+2+3+4+5… +n.

The above sequence consists of 1, 2 , 3, 4, 5… n as its terms. And as observed this sequence forms an arithmetic sequence with the first term as 1 and the common difference as +1.

The formula for the sum of the first n terms of an arithmetic sequence is

Using this formula, we can obtain a general expression of the 1st triangular sequence.

=

Therefore, the stage n or the nth triangular number is represented by the above equation.

For example, if we substitute n as 2, i.e the 2nd stage, it will be equal to

=

= 3

This holds true as we’ve already witnessed the 2nd stage having three dots.

Now let’s take 9th stage into consideration. It will be  =

= 45

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