Consider the theorem below. If a element R > 0 and n element Z > 0 then a^n - 1
ID: 2246569 • Letter: C
Question
Consider the theorem below. If a element R > 0 and n element Z > 0 then a^n - 1 = 1. Proof. We shall proceed by induction on n. If n = 1, then we have a^n - 1 = a^1 - 1 = a^0 - 1, establishing the base case. For the inductive step, let us assume that the theorem is true for 1, 2, n, and we shall prove that it holds for n + 1. We have a^(n + 1)-1 = a^n = a^n - 1 a^n - 1/a^(n - 1)-1 = (1) (1)/1 = 1. completing the induction and thus proving the theorem. The theorem is clearly false, so there must be an error in the proof. Find it.Explanation / Answer
proof consider a(n-1) -1 = 1 = a(n-1)
when n=2, we get a2-1 = a1 = (a1-1)*(a1-1)/a(1-1) -1 = 1/a-1 now if we consider the denominator to be 1 the for n=0;
an-1 = 1 which is false.