- #1
Proof and Math Induction Homework | Step-by-Step Solution
Click For Summary
Homework Help Overview
The discussion revolves around a proof involving mathematical induction, specifically focusing on the correct application of the induction method and the inequalities related to a sequence. Participants are exploring the nuances of proving statements for all integers greater than a certain value.
Discussion Character
- Exploratory, Conceptual clarification, Assumption checking
Approaches and Questions Raised
- Participants discuss the importance of correctly applying the induction hypothesis and the need to clarify the inequalities involved. There are suggestions to revisit foundational concepts of induction and to seek specific guidance on unclear aspects.
Discussion Status
Several participants have offered critical feedback and suggestions for improvement, indicating a productive exchange of ideas. There is recognition of the original poster's efforts and a general encouragement to deepen understanding of induction.
Contextual Notes
Some participants note the original poster's discomfort with the induction method and suggest reviewing textbook material for better comprehension. There is an acknowledgment of the original poster's previous lack of exposure to the subject matter.
- #2
epenguin
Science Advisor
- 3,638
- 1,014
It does not seem to me you are getting it correctly. If you are getting involved with ak+2 as well as ak+1 and ak I don't think you are going to get it.
I suggest you write out "The statement will be proved for n=(k+1) of we can prove that ...". That will guide your further steps.
I suggest you write out "The statement will be proved for n=(k+1) of we can prove that ...". That will guide your further steps.
- #3
phillyolly
- 157
- 0
Please expand your suggestion...
- #4
jgens
Gold Member
- 1,575
- 50
Why don't you try epenguin's suggestion first? Aside from the fact, based on the number of induction threads you've posted here today, you're clearly not comfortable with the method. I found that once I really understood the method intuitively, I became a lot more confident with my ability to write proofs by induction. So, is there anything that you find particularly discomforting or unclear about induction?
- #5
epenguin
Science Advisor
- 3,638
- 1,014
The next step is the whole idea of induction. Consult any examples you have done or followed.
In an induction proof a statement about a formula F(n) is true for n=k. What do you have to prove in order to be able to argue it is true for all n greater than k?
In an induction proof a statement about a formula F(n) is true for n=k. What do you have to prove in order to be able to argue it is true for all n greater than k?
- #6
╔(σ_σ)╝
- 839
- 2
You are not doing it correctly.
When you showed that this 1 \leq a_{n} \leq 2 inequality holds when n=2, how did you do it ?
You showed that
a_{2}= \frac{1}{2} + 1= 1.5 Correct ?
And you concluded that
1 \leq a_{2} \leq 2.
What you are not doing correctly is this same process for a_{k+1}
You have to show that
1 \leq a_{k+1} \leq 2
We know by inductive hypothesis that
1 \leq a_{k} \leq 2
From this we can tell that
\frac{1}{2} \leq \frac{a_{k}}{2} \leq 1
So what can you say about \frac{1}{a_{k}} ?
What inequality does it satisfy, if we know that 1 \leq a_{k} \leq 2 ?After that, what can we say about the question marks below.
? \leq\frac{a_{k}}{2}+ \frac{1}{a_{k}} \leq ?
When you showed that this 1 \leq a_{n} \leq 2 inequality holds when n=2, how did you do it ?
You showed that
a_{2}= \frac{1}{2} + 1= 1.5 Correct ?
And you concluded that
1 \leq a_{2} \leq 2.
What you are not doing correctly is this same process for a_{k+1}
You have to show that
1 \leq a_{k+1} \leq 2
We know by inductive hypothesis that
1 \leq a_{k} \leq 2
From this we can tell that
\frac{1}{2} \leq \frac{a_{k}}{2} \leq 1
So what can you say about \frac{1}{a_{k}} ?
What inequality does it satisfy, if we know that 1 \leq a_{k} \leq 2 ?After that, what can we say about the question marks below.
? \leq\frac{a_{k}}{2}+ \frac{1}{a_{k}} \leq ?
- #7
- #8
╔(σ_σ)╝
- 839
- 2
You have the right idea but you made a mistake in the inequality for \frac{1}{a_{n}}. Read your inequality and see if it makes sense.
- #9
epenguin
Science Advisor
- 3,638
- 1,014
In your first attempt you made a statement about ak+1 and ak, and you then went on to make another about ak+1 and ak and ak+2 which will get you nowhere.
In your second you leave out the statement about ak+1 and make one about only ak which will equally get you nowhere by itself. Try a happy medium! Better, please look up some other elementary example of an induction proof, because you do not (yet) have a problem with real analysis, you have a problem with induction. Which is a very easy idea - it might have applications or examples in real analysis which are not so easy, but you must get the easy part clear.
In your second you leave out the statement about ak+1 and make one about only ak which will equally get you nowhere by itself. Try a happy medium! Better, please look up some other elementary example of an induction proof, because you do not (yet) have a problem with real analysis, you have a problem with induction. Which is a very easy idea - it might have applications or examples in real analysis which are not so easy, but you must get the easy part clear.
- #10
╔(σ_σ)╝
- 839
- 2
Yes, i agree with the above post. You should seriously consider re-learning induction .
- #11
phillyolly
- 157
- 0
I see my mistake...
- #12
phillyolly
- 157
- 0
This is so new to me. I pass.
- #13
╔(σ_σ)╝
- 839
- 2
Can you re- read the chapter on induction in your txtbook and then ask specific questions about what you do not understand.
Similar threads
- · Replies 3 ·
- · Replies 15 ·
- · Replies 7 ·
- · Replies 3 ·
- · Replies 4 ·
- · Replies 1 ·