Number of Regions by dividing Circle by Chords

Theorem
Let $n$ points be marked on the circumference of a circle $C$.

Let chords be drawn between each pair of these points.

For each $n$, the maximum number $C \left({n}\right)$ of regions into which $C$ can be divided is as follows:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $C \left({n}\right)$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $31$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $57$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $99$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $163$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $256$
 * }
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $99$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $163$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $256$
 * }
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $256$
 * }

Proof

 * CircleDividedByChord1.png $\quad$ CircleDividedByChord2.png $\quad$ CircleDividedByChords3.png


 * CircleDividedByChords4.png $\quad$ CircleDividedByChords5.png $\quad$ CircleDividedByChords6.png