The topic we have chosen to investigate for our maths assignment is ‘sequences’. In order to do this we will firstly show a diagram of matchstick shapes increasing in size. Our first chosen stage is a triangle, in the second stage we will draw a table of results, the top line representing the orders of terms and the bottom line will represent the number of matchsticks used to make the sequence. From this table we will be able to see the ‘number increase’ in each step of the sequence.
With this information we hope to find the formula for the ‘nth term’. The ‘nth term’ is merely a representation of the order of sequence; therefore it can be substituted for any value in sequence. By using the formula we should be able to predict any number of the sequence. In order to find our formula we must have a consistent increase in our number of matchsticks. Since we have this we can look at number 4 in the order of sequence and compare it with the corresponding number of matchsticks. We see that the number of matchsticks is twice the size of number 4 plus 1.
This is written as 2n+1 Where n represents the order of sequence. The formula 2n allows us to work out any number in the sequence. Working out the difference in the number of matchsticks derives this formula (in this case the result is 2) we then multiply this difference by the order of sequence (this gives us 2n). To complete the formula we examine the 2n to find what has to be done with it to reach the accurate formula. These are achieved by substituting our information on the table into the formula and then determine what has to be added to the formula to make it work.
In this instance we will have to add 1 to make the formula work; i.e. 2xn 8+1=9 2×4=8 2n+1 is our formula. Triangles of Matchsticks As we can see a pattern has developed. The number of matchsticks is increasing by two in each set of triangles. We now put the order of sequence along with the number of matchsticks into a linear table. The 1st difference in not constant therefore its not linear, the 2nd difference is also not constant and that means it isn’t quadratic, but the 3rd difference is constant therefore the formula should take the form.
We are assuming this because of the previous sequence we did. In the 1st group of sequences the 1st difference was constant and the formula was in the form of N to the power of 1. In the second group of sequences the 1st difference was not constant so we looked at the 2nd difference and discovered that the 2nd difference was constant and that the formula took the form of n , or N to the power of 2. With this evidence we decided to look at a sequence which the 1st and 2nd where not constant but which had a constant 3rd difference.
If this sequence follows in the same pattern as the previous two, then it should have a formula in the form of n , or N to the power of 3. We can see the 3rd difference is constant and the result is number 6 therefore if the 3rd difference is 6 it takes the form of n . We will now double each number the number in the sequence to get a 3rd difference of 12 to see if the formula will take the form of 2n .