Screenshot of question attached. How exactly does (k^2 + 1)(a+b+c) change into k^2(a+b+c)+(a+b+c)^2 ?
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Answer:
At this point, probably everyone would be convinced that with 6 points there would be 32 regions, but it's not proved, it's just conjectured. The conjecture is that the number of regions when n points are connected is 2n -1.
Will finding the number of regions when there are six points on the circle prove the conjecture? No. If there are indeed 32 regions, all you have done is shown another example to support your conjecture. If there aren't 32 regions, then you have proved the conjecture wrong. In fact, if you go ahead and try the circle with six points on it, you'll find out that there aren't 32 regions.
You can never prove a conjecture is true by example.
You can prove a conjecture is false by finding a counter-example.
To prove a conjecture is true, you need some more formal methods of proof. One of these methods is the principle of mathematical induction.