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4. More weak induction Use weak induction on n to prove the following claim: n! < n" for all integers n >1

4. More weak induction Use weak induction on n to prove the following claim: n! < n" for all integers 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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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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