# Maths Internal Assessment -triangular and stellar numbers

Maths Internal Assessment

## Term 2 2011

Question 1:

Complete the triangular numbers sequence with three more terms.

The table below summarises the data (above) into a table so it is easier to analyse.

Question 2:

Find a general statement that represents the nth triangular number in terms of n.

The table below shows the connection and calculation between the term and the number of dots for the given triangular numbers.

Square numbers are also geometric shapes which lead to special numbers. Therefore, we must take into account these square numbers in order to find out the general statement.

When we compare the square numbers to the triangular numbers, a connection is discovered. We can see that:

Another method in finding the general statement is by using multiplication. By this I mean:

Therefore, by multiplying one term and the term next to it, then dividing the multiplication by 2, we can determine the number of dots in each triangle.

Yet another method in determining this general statement is by thinking logically. Technically, a triangle is half a square. Therefore, as the number of dots in a square number is n2, the equation should be n2/2. Except we must take into account the dots that we cut in half and include them in the equation. The dots that we cut in half will be represented as n/2. Therefore the statement is:

Question 3:

Find the number of dots (i.e. the stellar numbers) in each stage up to S6. Organize the data so that you can recognize and describe any patterns.

The table below organizes the data so that we can recognise and describe the patterns.

Excluding the first layer (layer 1), as the stage number (Sn) and layer increases by 1, the number of dots correlatively increase by 12.

Question 4:

Find an expression for the 6-stellar number at S7.

As we discovered the connection between the number of layers and the number of dots, an expression for S7 can be calculated.

Therefore, the expression for the 6-stellar number at stage S7 is (12 x 21) +1.

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