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Theorem 5.1.1. For all n E N, 1+2+3+...+n = 2 m(a+1) (5.1) The following sentences are enumerated. Put them in order as presented in the lesson material to create a proof of Theorem 5.1.1. that uses an induction argument. Enter your answer as a 6 digit integer. (For example, 123456 is in a valid format for an answer, but it is incorrect.) 1. Inductive step: Assume that P(n) is true, that is equation (5.1) holds for some nonnegative integer n. 2. So it follows by induction that P(n) is true for all nonnegative n . 3. We use induction. 4. Base case: P(0) is true, because both sides of equation (5.1) equal zero when = 0. 5. The induction hypothesis P(n) will be equation (5.1). 6. Then adding n +1 to both sides of the equation implies that n(n + 1) 1+2+3+ +n+ (n +1) : + (n + 1) 2 (n + 1)(n + 2) (by simple algebra) which proves P(n + 1). ||

Theorem 5.1.1. For all n E N, 1+2+3+...+n = 2 m(a+1) (5.1) The following sentences are enumerated. Put them in order as presented in the lesson material to create a proof of Theorem 5.1.1. that uses an induction argument. Enter your answer as a 6 digit integer. (For example, 123456 is in a valid format for an answer, but it is incorrect.) 1. Inductive step: Assume that P(n) is true, that is equation (5.1) holds for some nonnegative integer n. 2. So it follows by induction that P(n) is true for all nonnegative n . 3. We use induction. 4. Base case: P(0) is true, because both sides of equation (5.1) equal zero when = 0. 5. The induction hypothesis P(n) will be equation (5.1). 6. Then adding n +1 to both sides of the equation implies that n(n + 1) 1+2+3+ +n+ (n +1) : + (n + 1) 2 (n + 1)(n + 2) (by simple algebra) which proves P(n + 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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Theorem 5.1.1. For all n E N, n(n+1) 1+2+3+ +n = (5.1) .. The following sentences are enumerated. Put them in order as presented in the lesson material to create a proof of Theorem 5.1.1. that uses an induction argument. Enter your answer as a 6 digit integer. (For example, 123456 is in a valid format for an answer, but it is incorrect.) 1. Inductive step: Assume that P(n) is true, that is equation (5.1) holds for some nonnegative integer n. 2. So it follows by induction that P(n) is true for all nonnegative n. 3. We use induction. 4. Base case: P(0) is true, because both sides of equation (5.1) equal zero when = 0. 5. The induction hypothesis P(n) will be equation (5.1). 6. Then adding n+1 to both sides of the equation implies that п(п + 1) 1+2+ 3+ ·..+n+ (n + 1) + (n + 1) 2 (n + 1)(n + 2) (by simple algebra) 2 which proves P(n +1).
Transcribed Image Text:Theorem 5.1.1. For all n E N, n(n+1) 1+2+3+ +n = (5.1) .. The following sentences are enumerated. Put them in order as presented in the lesson material to create a proof of Theorem 5.1.1. that uses an induction argument. Enter your answer as a 6 digit integer. (For example, 123456 is in a valid format for an answer, but it is incorrect.) 1. Inductive step: Assume that P(n) is true, that is equation (5.1) holds for some nonnegative integer n. 2. So it follows by induction that P(n) is true for all nonnegative n. 3. We use induction. 4. Base case: P(0) is true, because both sides of equation (5.1) equal zero when = 0. 5. The induction hypothesis P(n) will be equation (5.1). 6. Then adding n+1 to both sides of the equation implies that п(п + 1) 1+2+ 3+ ·..+n+ (n + 1) + (n + 1) 2 (n + 1)(n + 2) (by simple algebra) 2 which proves P(n +1).
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