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Prove the following statement by mathematical induction. n + 1 For every integer n 2 0, Si.2' =n2n + 2+ 2. %3D i = 1

Prove the following statement by mathematical induction. n + 1 For every integer n 2 0, Si.2' =n2n + 2+ 2. %3D 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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Prove the following statement by mathematical induction. For every integer n 2 0, n+1 i- 2' = n· 2n + 2+2. i = 1 Proof (by mathematical induction): Let P(n) be the equation n +1 si.2' = n · 2n + 2 + 2. i = 1 We will show that P(n) is true for every integer n > 0. Show that P(0) is true: Select P(0) from the choices below. 0 + 1 1. 21 = 1 · 21 + 2 + 2 i = 0 0 + 1 OS i.2' = 0 · 20 + 2 + 2 i = 1 n+1 O5 i.2' = 0 - 20 + 2 + 2 i = 1 O 2 = 0 · 20 + 2+ 2 The 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+1 i. 2k i = 1 k+1 i = 1 i. 2k + 1 i = 1 k+1 The right-hand side of P(k) is EThe inductivo bvoothocic ctatect the twe cidos of DA are cauall EME IME IME ME
Transcribed Image Text:Prove the following statement by mathematical induction. For every integer n 2 0, n+1 i- 2' = n· 2n + 2+2. i = 1 Proof (by mathematical induction): Let P(n) be the equation n +1 si.2' = n · 2n + 2 + 2. i = 1 We will show that P(n) is true for every integer n > 0. Show that P(0) is true: Select P(0) from the choices below. 0 + 1 1. 21 = 1 · 21 + 2 + 2 i = 0 0 + 1 OS i.2' = 0 · 20 + 2 + 2 i = 1 n+1 O5 i.2' = 0 - 20 + 2 + 2 i = 1 O 2 = 0 · 20 + 2+ 2 The 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+1 i. 2k i = 1 k+1 i = 1 i. 2k + 1 i = 1 k+1 The right-hand side of P(k) is EThe inductivo bvoothocic ctatect the twe cidos of DA are cauall EME IME IME ME
[The inductive hypothesis states that the two sides of P(k) are equal.] (k+1)+1 (k1)+1 (k+1)+1 We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i. 2'. when the final term of S i - 2' is written separately, the result is i. 2' = The right-hand side of P(k + 1) is After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes + (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.] Need Help? Read It
Transcribed Image Text:[The inductive hypothesis states that the two sides of P(k) are equal.] (k+1)+1 (k1)+1 (k+1)+1 We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i. 2'. when the final term of S i - 2' is written separately, the result is i. 2' = The right-hand side of P(k + 1) is After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes + (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.] Need Help? Read It
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