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Given that P(n) is the equation 1+3+5+7++ (2n − 1) = n², where n is an integer such that n ≥ 1, we will prove that P(n) is true for all n ≥ 1 by induction. (a) Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k≥ 1 be a natural number. Assume that P(k) is true. Write P(k). (c) Inductive step: i. Write P(k+ 1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. Justify all of your steps.

Given that P(n) is the equation 1+3+5+7++ (2n − 1) = n², where n is an integer such that n ≥ 1, we will prove that P(n) is true for all n ≥ 1 by induction. (a) Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k≥ 1 be a natural number. Assume that P(k) is true. Write P(k). (c) Inductive step: i. Write P(k+ 1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. Justify all of your steps.

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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## Mathematical Induction Proof Given that \( P(n) \) is the equation \( 1 + 3 + 5 + 7 + \ldots + (2n - 1) = n^2 \), where \( n \) is an integer such that \( n \geq 1 \), we will prove that \( P(n) \) is true for all \( n \geq 1 \) by induction. ### (a) Base case: i. Write \( P(1) \). ii. Show that \( P(1) \) is true. *In this case, this requires showing that a left-hand side is equal to a right-hand side.* ### (b) Inductive hypothesis: Let \( k \geq 1 \) be a natural number. Assume that \( P(k) \) is true. Write \( P(k) \). ### (c) Inductive step: i. Write \( P(k + 1) \). ii. Use the assumption that \( P(k) \) is true to **prove that \( P(k + 1) \) is true**. Justify all of your steps.
Transcribed Image Text:## Mathematical Induction Proof Given that \( P(n) \) is the equation \( 1 + 3 + 5 + 7 + \ldots + (2n - 1) = n^2 \), where \( n \) is an integer such that \( n \geq 1 \), we will prove that \( P(n) \) is true for all \( n \geq 1 \) by induction. ### (a) Base case: i. Write \( P(1) \). ii. Show that \( P(1) \) is true. *In this case, this requires showing that a left-hand side is equal to a right-hand side.* ### (b) Inductive hypothesis: Let \( k \geq 1 \) be a natural number. Assume that \( P(k) \) is true. Write \( P(k) \). ### (c) Inductive step: i. Write \( P(k + 1) \). ii. Use the assumption that \( P(k) \) is true to **prove that \( P(k + 1) \) is true**. Justify all of your steps.
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