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2.12 L) Special Sequences: Triangle, Square & Cube
2.12 L) Special Sequences: Triangle, Square & Cube
There are a few special sequences that may come up in the exam. We are going to be looking at these special sequences in this section. It is definitely worth making some notes about each of the sequences on a revision card.
Triangle Numbers
The triangle numbers are obtained by drawing out a triangle with dots and adding an extra row each time. The triangle numbers are:
The triangle numbers are obtained by drawing out a triangle with dots and adding an extra row each time. The triangle numbers are:
The first term is 1. We then add another row, which is 2 more dots. This means that the triangle is made up of 3 dots.
We obtain the third term by adding another row to the second triangle. When we do this, we add 3 more dots, and this means that the triangle is now made up of 6 dots.
We add another row for the fourth triangle, which means that the fourth triangle is made out of 10 dots.
We are also able to use the term to term rule for the triangle numbers, which is to add one more than we previously added each time. We add 2 to the first term to obtain the second term. We add 3 to the second term to obtain the third term. We add 4 to the third term to obtain the fourth term, etc.
The triangular numbers can be found by using the following nth term formula:
Let’s check that this formula is the correct formula and we will do this by checking the fourth term. The value of n for the fourth term is 4 and we are going to sub 4 into the above equation.
This tells us that the fourth triangular number is 10, which is what we found out earlier.
Let’s use the nth term formula to find what the 22nd term would be. We can find the 22nd term by subbing n in as 22.
The 22nd term in the sequence is 253.
Square & Cubic Numbers
You may also be asked about square and cube number sequences in the exam.
Here are the first 10 terms in the square number sequence:
Here are the first 5 terms for the cube number sequence: