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Prove the following statement by mathematical induction. n+ 1 For every integer n2 0, i I.2' = n- 2" *2+ 2. i = 1

Prove the following statement by mathematical induction. n+ 1 For every integer n2 0, i I.2' = n- 2" *2+ 2. i = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please read the question and leave notes on the answer where appropriate. Please DO NOT skip any steps. Please double check your answer. DO NOT SUBMIT A TYPED RESPONSE. Please use a hand written response that is LEGABLE. Please review your answer to make sure all steps are clearly visable. Please double check your work. Thank you. 

Can you please explain what the right side should be to prove this statement? I'm very confused by the two variables. 

 

Prove the following statement by mathematical induction. n+1 For every integernz 0, Fi.2' = n: 2" + 2+ 2. i = 1 Proof (by mathematical induction): Let P(n) be the equation n+1 Si:2' = n 2' + 2. i= 1 We 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 + 1 - 21 +2 + 1. i = 0 2 O 2 = 0- 2" + 2 + 2 0 + 1 Oi: 2 = 0 - 20 + 2+ 2 i = 1 n + 1 Ei.2' = 0. 20 + 2+ 2 i = 1 The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 0, if P(k) is true, then P(k + 1) is true: Let k be any integer with k z 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below. k+1 i = 1 k+1 i = 1 k+1 Σ i= 1 k+1 The right-hand side of P(k) is (i+1) · 2(?+1)+2+2 IThe inductive hvpothesis states that the two sides of P(k) are equal.1
Transcribed Image Text:Prove the following statement by mathematical induction. n+1 For every integernz 0, Fi.2' = n: 2" + 2+ 2. i = 1 Proof (by mathematical induction): Let P(n) be the equation n+1 Si:2' = n 2' + 2. i= 1 We 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 + 1 - 21 +2 + 1. i = 0 2 O 2 = 0- 2" + 2 + 2 0 + 1 Oi: 2 = 0 - 20 + 2+ 2 i = 1 n + 1 Ei.2' = 0. 20 + 2+ 2 i = 1 The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 0, if P(k) is true, then P(k + 1) is true: Let k be any integer with k z 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below. k+1 i = 1 k+1 i = 1 k+1 Σ i= 1 k+1 The right-hand side of P(k) is (i+1) · 2(?+1)+2+2 IThe inductive hvpothesis states that the two sides of P(k) are equal.1
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