Final answer:
By the principle of mathematical induction, it can be shown that n lines, with no two parallel and no three concurrent, separate the plane into (n^2 + n + 2)/2 regions by starting with a base case and assuming the formula holds for n lines, then proving it holds for n+1.
Step-by-step explanation:
To demonstrate that n lines separate the plane into (n^2 + n + 2)/2 regions under the conditions that no two of these lines are parallel and no three pass through a common point, we can use a method of induction.
Firstly, we establish our base case: for n=1, one line divides the plane into 2 regions which satisfies our formula as (1^2 + 1 + 2)/2 = 2. Now, assuming that the formula is true for n lines, we shall add one more line. This new line (the (n+1)th line) will intersect with all the previous n lines since none of them are parallel or concurrent. Each intersection creates a new region.
The n lines created n(n+1)/2 regions, as per our formula. Now, the (n+1)th line intersects the n previous ones in n points, therefore adding n new regions. This gives us a total of (n(n+1)/2) + n = (n^2 + n + 2 + 2n)/2 = ((n+1)^2 + (n+1) + 2)/2 regions. By the inductive step, we have shown our formula holds for n+1 lines if it holds for n lines.
Since it holds for our base case, it holds for all positive integers n by the principle of mathematical induction, fulfilling the conditions stated in the problem.