Prove the following statement by mathematical induction.n+1For every integer n 2 0, ri:2' = n · 2" + 2 + 2.i = 1Proof (by mathematical induction): Let P(n) be the equationi. 2' = n. 2" + 2 + 2.i = 1We will show that P(n) is true for every integer n 2 0.Show that P(0) is true: Select P(0) from the choices below.0+1Fi. 2' = 0 - 2° + 2 + 2i= 1O 2 = 0 . 27 + 2 + 2F1.2' = 1 - 21 + 2 +i = 0i: 2' = 0. 20 + 2 + 2The selected statement is true because both sides of the equation equal the same quantity.Show that for each integer k > 0, if P(k) is true, then P(k + 1) is true:Let k be any integer with k 2 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.k+1i = 1k+1i. 2kk 2i. 2k +The right-hand side of P(k) is 1[The inductive hypothesis states that the two sides of P(k) are equal.](k+1)+1We must show that P(k + 1) is true. The left-hand side of P(k + 1) is S(k+1)+1(k+1)+1i. 2'. When the final term ofi- 2' is written separately, the result isi- 2' =i: 2' +. The right-hand side of P(k + 1) isAfter substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomesi = 1i = 1i = 1i = 1+ (k:+ 2)2k + 2 ). when the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal+ 2. 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.]

Name: Prove the following statement by mathematical induction.n+1For every
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Prove the following statement by mathematical induction.n+1For every integer n 2 0, ri:2' = n · 2" + 2 + 2.i = 1Proof (by mathematical induction): Let P(n) be the equationi. 2' = n. 2" + 2 + 2.i = 1We will show that P(n) is true for every integer n