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Weak induction in steps Let P(n) be the following statement: 1 n i(i+1) n+1 The following sub-problems guide you through a proof by weak induction that P(n) holds for all n EZt. (a) In order to understand the claim, verify it by hand for n = 4. (b) the base case. What is the statement P(1)? Show that P(1) is true, which completes (c) What is the inductive hypothesis? (d) What do you need to prove in the inductive step? (e) Complete the inductive step.

Weak induction in steps Let P(n) be the following statement: 1 n i(i+1) n+1 The following sub-problems guide you through a proof by weak induction that P(n) holds for all n EZt. (a) In order to understand the claim, verify it by hand for n = 4. (b) the base case. What is the statement P(1)? Show that P(1) is true, which completes (c) What is the inductive hypothesis? (d) What do you need to prove in the inductive step? (e) Complete the inductive step.

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Sequences And Series
Section: Chapter Questions
Problem 4CC
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1. Sequence Consider the sequence {a,}nEN where an = 2" – 3". (a) Write the first five terms of the sequence. (b) State whether the sequence appears to be (non-)increasing or (non-)decreasing. (c) Is the sequence an arithmetic progression? Explain. (d) Is the sequence a geometric progression? Explain. 2. Summation (a) Write the sum 50 + 51+52+ ..+ 99 + 100 in summation notation. (b) Compute the value this summation. 3. Weak induction in steps Let P(n) be the following statement: 1 Σ n i(i+1) i=1 n +1 The following sub-problems guide you through a proof by weak induction that P(n) holds for all ne Zt. (a) In order to understand the claim, verify it by hand for n = 4. (b) the base case. What is the statement P(1)? Show that P(1) is true, which completes (c) What is the inductive hypothesis? (d) What do you need to prove in the inductive step? (e) Complete the inductive step. 4. More weak induction Use weak induction on n to prove the following claim: n! < n" for all integers n >1
Transcribed Image Text:1. Sequence Consider the sequence {a,}nEN where an = 2" – 3". (a) Write the first five terms of the sequence. (b) State whether the sequence appears to be (non-)increasing or (non-)decreasing. (c) Is the sequence an arithmetic progression? Explain. (d) Is the sequence a geometric progression? Explain. 2. Summation (a) Write the sum 50 + 51+52+ ..+ 99 + 100 in summation notation. (b) Compute the value this summation. 3. Weak induction in steps Let P(n) be the following statement: 1 Σ n i(i+1) i=1 n +1 The following sub-problems guide you through a proof by weak induction that P(n) holds for all ne Zt. (a) In order to understand the claim, verify it by hand for n = 4. (b) the base case. What is the statement P(1)? Show that P(1) is true, which completes (c) What is the inductive hypothesis? (d) What do you need to prove in the inductive step? (e) Complete the inductive step. 4. More weak induction Use weak induction on n to prove the following claim: n! < n" for all integers n >1
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