Use mathematical induction to prove the following statement. For every integer n ≥ 2, (1-2) (1-3)... (1-2)= n + 1 Proof (by mathematical induction): Let the property P(n) be the equation (1 - 237) (1-33) ··· (1 - 1327) We will show that P(n) is true for every integer n ≥ n+1 2n 2n +1 Show that P is true: Before simplification, the left-hand side of is 1- and the right-hand side is After simplification, both sides can be shown to equal 2.2 Show that for each integer k ≥ if P(k) is true, then P 1 (1-2) (1-3) (1-2) ... (1- and the right-hand side of P(k) is 2k [The inductive hypothesis is that the two sides of P(k) are equal.] We must show that Pl (1-2) (1-3) (1-2) ... is true: Let k be any integer with k ≥ 2, and suppose that P(k) is true. Before any simplification, the left-hand side of P(k) is is true. In other words, we must show that the left- and right-hand sides of P are equal. The left-hand side of P and the next-to-last factor in the left-hand side is 1 (1-12) So, when the next-to-last factor is explicitly included in the expression for the left-hand side, the result is 1 1- 1 (-) (-)(-)-(-)(( After substitution from the inductive hypothesis, the left-hand side of P 2k Before simplification, the right-hand side of P When the left- and right-hand sides of P Hence, P 2k 2k 2k(k+1) IS +1 1 becomes 2(k + 1) are simplified, both can be shown to equal is true, which completes the inductive step. Thus, P is true.