3n2-(3n-1)
We got this by first collecting the data from the cable sizes, up to cable size 5, by counting them and drawing them. We then took them in to nth terms as follows:
Size 1 2 3 4 5
No. of Wires 1 7 19 37 61
1st term +6 +12 +18 +24
2nd term +6 +6 +6
So we realised that it keeps increasing on the second term, meaning it's a square number, therefore, the first bit of eqution was 3n2. However when we did 3n2 what happened was
Size 1 2 3 4 5
No. of wires 3 12 27 48 75
We knew by this that something was being subtracted. To find out this, we compared the two sets of numbers to find the difference
1st set 1 7 19 37 61
Difference +2 +5 +8 +11 +14
2nd set 3 12 27 48 75
After that, we had a look at the differences
Differnce 2 5 8 11 14
1st term +3 +3 +3 +3
As this wasn't the second term, the equation is 3n. However, because with 3n you get;
3 6 9 12 15
There must be a difference, which is -1. Therefore the equation would be 3n-1.
When we put these two together, we get 3n2-3n-1. However, the problem is that when we use, let's say size 5, what happens is:
3x5(squared)-3x5-1=59
What this made use realise is that we need to put the subtraction in brackets for it to work making it 3n2-(3n-1) which works, as the example of size 5 again:
3x5(squared)-(3x5-1)=75-14=61