Use mathematical induction to prove the following statement. n+1 : 2, (¹ - 2 ) (¹ - ¹3 ) --- (¹ - 22 ) -^2¹ Proof (by mathematical induction): Let the property P(n) be the equation (₁-21) (¹ - 1) (¹ - 1) - We will show that P(n) is true for every integer n 2 For every integer n 2 2, Show that P Show that for each integer k is true: Before simplification, the left-hand side of Pis (¹-2)(¹-3) (¹ - 1) and the right-hand side of P(k) is We must show that |- (¹ - 2 ) ) ( ¹ - 3 - ) ( ¹ - - - - - - - (¹ - 1- n+1 2n if P(k) is true, then P 2k [The inductive hypothesis is that the two sides of P(k) are equal.] PC When the left- and right-hand sides of P 1 +1 Before simplification, the right-hand side of P and the next-to-last factor in the left-hand side is (1-21)- (¹ - 2 ) (¹ - ² ) (¹ - - ) --- (1 - 2 ) ( ¹ - ( 1 is true. In other words, we must show that the left and right-hand sides of P After substitution from the inductive hypothesis, the left-hand side of P I).(+-+ T)-( 1 2k -O). 2k(k+1) 1- 2(k+1) So, when the next-to-last factor is explicitly included in the expression for the left-hand side, the result is 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 and the right-hand side is becomes 1 +1. After simplification, both sides can be shown to equal 2.2 are simplified, both can be shown to equal are equal. The left-hand side of . Thus, P is true.