EXAMPLE: We already discovered that for certain types of patterns, where the number advances by the same amount every time, that the function rule for this is jx + c where the coefficient j is the size of the step and the constant c is the amount needed to adjust the value to work for the first term. For instance:
Term n | 1 | 2 | 3 | 4 | 5 | 6 | ... | n | ... | 200 |
Function f(n) | 8 | 11 | 14 | 17 | 20 | 23 | ... | ... |
The function is moving up by 3 each time so the function rule is 3n + something.
For term 1, f(1)=8, which means 3(1) + c = 8, which means c is 5.
Completing the table below looks like this:
Term n | 1 | 2 | 3 | 4 | 5 | 6 | ... | n | ... | 200 |
Function f(n) | 8 | 11 | 14 | 17 | 20 | 23 | ... | 3n+5 | ... | 605 |
For an n-sided figure, the formula for the number of total diagonals is f(n) = n(n-3)/2 .If n is the number of vertices, then n-3 is the number of diagonals out of a single vertex. The number of diagonals might seem to be n(n-3) [number of vertices times diagonals per vertex] but this ends up counting each vertex twice. Thus we divide this number by 2 and we get our formula f(n) = n(n-3)/2. [For a handshake it was n(n-1)/2 - would include the sides!]