Please hlep Prove the following statement by mathematical induction.1For every integer n 2 1,1.21112-33-4n(n + 1)n +1Proof (by mathematical induction): Let P(n) be the equation11111.22-33-4n(n + 1)n+1We will show that P(n) is true for every integer n2 1.Show that P(1) is true: Select P(1) from the choices below.1O P(1)1-21O P(1)1+111.211111-22 -33. 41-21+ 111%3D1.21(1 + 1)1+1The selected statement is true because both sides of the equation equal the same quantity.Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true:Let k be any integer with k 2 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.111- 2k(k + 1)1111- 22-33.411111- 22-33. 4k(k + 1)1111.22-33. 4k(k + 1)1The right-hand side of P(k) is*(k +1)[The inductive hypothesis states that the two sides of P(k) are egual.] [The inductive hypothesis states that the two sides of P(k) are equal.]We must show that P(k + 1) is true. P(k + 1) is the equation*+111(&+1) +11.22-33- 4*+1k+1Which of the following choices shows the result of applying the inductive hypothesis to the expression on the left-hand side of P(k + 1)?k1k+1(k + 1)(k + 2)k1k +1k(k + 1)(k + 1)(k + 2)1-2 K(k + 1)(k + 1)(k + 2)11111.22-33. 4k(k + 1)(k + 1)(k + 2)k +1( +1) +1When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal. Hence P(k + 1) is true, which completes the inductive step.[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

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Prove the following statement by mathematical induction. 1 1 1 1 For every integer n2 1, 1-2 2-3 3-4 n(n + 1) n +1 Proof (by mathematical induction): Let P(n) be the equation 1 1 1 1 1-2 2-3 3- 4 n(n + 1) n+1 We will show that P(n) is true for every integer n2 1. Show that P(1) is true: Select P(1) from the choices below. 1 O P(1) 1-2 1 O P(1) 1 +1 1 1-2 1 1 1 1 1-2 2-3 3. 4 1-2 1 +1 1 1 1- 2 1(1 + 1) 1+1 The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choic 1 1 1- 2 k(k + 1) 1 1 1.2 2-3 3. 4 1 1 1 1-2 2-3 3. 4 k(k + 1) 1 1 1 2 -3 k(k + 1) 1-2 3.4 1 The right-hand side of P(k) is *(k +1) [The inductive hypothesis states that the two sides of P(k) are egual.]