Inequality Proof Homework: Proving 2^k+1 $\geq$ 1/2

In summary, the problem is trying to prove that the sum of the terms in a sequence is greater than the sum of the terms in the sequence plus one.
  • #1
jeff1evesque
312
0

Homework Statement


[tex]\frac{1}{2^{k}+1} + \frac{1}{2^{k} +2} + ... + \frac{1}{2^k + 2^k} \geq \frac{1}{2}[/tex]
2. The attempt at a solution
Not too sure, I am working on a larger proof (not too much difficult) and the above is my attempt to prove the induction step k+1 (since [tex]\frac{1}{2^k + 2^k} = \frac{1}{2^{k+1}}[/tex]).

Should i try to factor out [tex]\frac{1}{2^k}[/tex]?
 
Physics news on Phys.org
  • #2
Hi jeff! :smile:

Hint: they're all greater than the last one. :wink:
 
  • #3
tiny-tim said:
Hi jeff! :smile:

Hint: they're all greater than the last one. :wink:

I still don't see it, i wish it were clear to me.

Thanks though,

Jeffrey Levesque
 
  • #4
jeff1evesque said:
I still don't see it, i wish it were clear to me.

Thanks though,

Jeffrey Levesque

If a < b < c
a + a + a < a + b + c

Right?

Try something along those lines.
 
  • #5
jeff1evesque said:
I still don't see it, i wish it were clear to me.

Thanks though,

Jeffrey Levesque

Don't you agree with tiny-tim that the last term in the sum is smaller than the rest? How many terms in the sequence are there?
 
  • #6
Dick said:
Don't you agree with tiny-tim that the last term in the sum is smaller than the rest? How many terms in the sequence are there?

Yes I agree, but that isn't helping me at all- I just don't know how to formulate a proof for this.
 
  • #7
There are [tex]2^{k+1}[/tex] terms
 
  • #8
jeff1evesque said:
There are [tex]2^{k+1}[/tex] terms

Are you sure? Count again. : )
 
  • #9
l'Hôpital said:
Are you sure? Count again. : )

[tex]2^k[/tex] terms.
 
  • #10
never minnd, I actually did this in a different problem- thanks everyone
 
  • #11
I also noticed that the last term on the LHS is the smallest, but I was confused for awhile since I thought the denominators were 2^k + 2^0, 2^k + 2^1, ..., 2^k + 2^k, in which case there are k+1 terms, and the estimate fails.

Anyways assuming the progression is what everyone else thinks it is, then basically [itex]2^i \leq 2^k[/itex] for i = 1, 2, ..., k so
[tex]2^k + 2^i \leq 2^k + 2^k = 2^{k+1} \Rightarrow \frac{1}{2^k + 2^i} \geq \frac{1}{2^{k+1}} [/tex]
for i = 1, 2, ..., k.
 

1. How do I begin solving this inequality proof homework?

To begin solving this inequality proof homework, you need to understand the properties of inequalities and how to manipulate them. Start by identifying the given inequality and the desired outcome. Then, use algebraic techniques such as factoring, substitution, or logarithms to manipulate the inequality until the desired outcome is reached.

2. What is the significance of the inequality 2^k+1 $\geq$ 1/2?

The inequality 2^k+1 $\geq$ 1/2 is significant because it represents a key concept in the study of inequalities – the power rule. This rule states that when an inequality contains a variable raised to a power, the inequality is preserved when the power is applied to both sides. In this case, the inequality 2^k+1 $\geq$ 1/2 can be manipulated using the power rule to prove the desired outcome.

3. Can I use any algebraic technique to solve this inequality proof homework?

Yes, you can use any algebraic technique to solve this inequality proof homework as long as it follows the rules of algebra. This includes techniques such as factoring, substitution, logarithms, and more. Just make sure that your manipulations are valid and do not change the original inequality.

4. Are there any common mistakes to avoid when solving this inequality proof homework?

Yes, there are a few common mistakes to avoid when solving this inequality proof homework. One mistake is incorrectly applying the power rule and changing the original inequality. Another is making algebraic errors, such as forgetting to distribute or simplifying incorrectly. It is important to double-check your work and make sure all steps are valid.

5. How can I check if my solution to this inequality proof homework is correct?

You can check if your solution to this inequality proof homework is correct by substituting your solution back into the original inequality and simplifying. If your solution satisfies the inequality, then it is correct. You can also check your work by using a graphing calculator or an online inequality solver to see if the graph of your solution matches the graph of the original inequality.

Similar threads

Replies
12
Views
813
  • Calculus and Beyond Homework Help
Replies
2
Views
454
  • Calculus and Beyond Homework Help
Replies
9
Views
846
  • Calculus and Beyond Homework Help
Replies
1
Views
220
  • Calculus and Beyond Homework Help
Replies
8
Views
78
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
965
  • Calculus and Beyond Homework Help
Replies
3
Views
482
  • Calculus and Beyond Homework Help
Replies
1
Views
546
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
Back
Top