How do I create an explicit equation for this? What is the maximum number of regions into which 1 chord can divide a circle? 2 chords? 3 chords? 20 chords? Find an explicit equation relating the number of regions to the number of chords, n.
the formula for an explicit equation: f(n)=f(1)+d(n-1)
the table created from this information shows a quadratic relationship so how would i create an explicit equation? 
I know there is a method called the brute force method but i do not know how to use it
 A: This is a nice result sometimes known as the "pancake cutting problem" or "lazy caterer problem". Let $k$ denote the number of cuts and $f$ be the function that maps a number of cuts $k$ to the corresponding maximum number of regions created by performing $k$ cuts $f(k)$. 
When in doubt it's useful try a first few examples. Getting our hands dirty we find
\begin{align*}
  f(1) &= 2 \\
  f(2) &= 4 \\
  &= 2 + f(1) \\
  f(3) &= 7 \\
  &= 3 + f(2) \\
  &= 3 + 2 + f(1) \\
  f(4) &= 14 = 4 + f(3) \\
  &= 4 + 3 + 2 + f(1)
\end{align*}
We're starting to get the sneaking suspicion that the general rule for $k$ cuts can be given by
\begin{equation*}
  f(k) = k + f(k - 1)
\end{equation*}
To see that this is indeed the case it's useful to sketch a few of our examples so that we may build a geometric argument. Suppose we have already have $k$ cuts diving our circle (in any way, not necessarily into a maximal number of regions). The next $(k + 1)th$ cut will at most intersect the previous $k$ cuts, each at a single point. Every time the $(k + 1)$th cut intersects an older cut it splits the region in two generating an additional region. Before reaching the other side of the circle the $(k + 1)$th cut will have intersected $k$ lines and so the $(k + 1)th$ cut will have created another $k$ regions. When the $(k + 1)th$ cut finally reaches the other side of the circle it will have split this final region in two and create another region. That is, the $(k + 1)$ will have created $k + 1$ new regions in total.
Constructing that argument was the hard part. We can now use some algebra to arrive to a solution.
\begin{align*}
  f(k) &= k + f(k - 1) \\
&= \sum^k_{l = 2} l + f(1), \quad k \geq 2 \\
&= \sum^k_{l = 2} l + 2 \\
&= \sum^k_{l = 1} l + 1 \\
&= \frac{k(k + 1)}{2} + 1 \\
&= \frac{k^2 + k + 2}{2}
\end{align*}
where the second to last step is a well-known result (partial sum of the first $k$ integers).
A: Once you know it is quadratic, you have $f(n)=an^2+bn+c$.  Compute $f(1), f(2), f(3)$.  Now you can write $f(3)=a\cdot 3^2+b\cdot 3+c$ and two similar equations.  Solve them simultaneously for $a,b,c$
