Can AM-GM inequality prove 2^n > 1+n\sqrt{2^{n-1}} for positive integers?

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In summary, The problem is to prove that for any positive integer n, 2^n is greater than or equal to 1 + n√2^(n-1). The attempted solution suggested using the AM GM HM inequality and trying different numbers, but it was pointed out that this may not be correct for all positive integers. The conversation then turned to using mathematical induction, which is a recommended method for solving problems of this type. Another suggestion was to use the binomial theorem on (1+1)^n to get closer to the desired inequality. Finally, a hint was given to use the fact that 2^n = 2 x 2^(n-1) and then apply the AM GM inequality.
  • #1
utkarshakash
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Homework Statement


If n is a positive integer, prove that [itex]2^n > 1+n\sqrt{2^{n-1}}[/itex]

Homework Equations



The Attempt at a Solution


I am thinking of applying AM GM HM inequality. But which numbers should I take to arrive at this inequality?
 
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  • #2
utkarshakash said:

Homework Statement


If n is a positive integer, prove that [itex]2^n > 1+n\sqrt{2^{n-1}}[/itex]

Homework Equations



The Attempt at a Solution


I am thinking of applying AM GM HM inequality. But which numbers should I take to arrive at this inequality?

...This may not be correct for any number n that is a positive integer.

For example, [itex]2^{1}[/itex] is not greater than [itex]1 + n\sqrt{2^{1-1}}[/itex]. In fact, they are equivalent.
 
  • #3
Mandelbroth said:
...This may not be correct for any number n that is a positive integer.

For example, [itex]2^{1}[/itex] is not greater than [itex]1 + n\sqrt{2^{1-1}}[/itex]. In fact, they are equivalent.

Ok so assume that it is 'greater that or equal to' instead of just 'is greater than' and prove it
 
  • #4
utkarshakash said:
Ok so assume that it is 'greater that or equal to' instead of just 'is greater than' and prove it

No: YOU prove it, or at least show some effort towards the solution. Read the Forum rules!

RGV
 
  • #5
Ray Vickson said:
No: YOU prove it, or at least show some effort towards the solution. Read the Forum rules!

RGV

Hey I really don't know how to solve this. I need some hints to get started. I've already stated that I am thinking of solving it using AM GM HM inequality. I know nothing more than this.
 
  • #6
utkarshakash said:
Hey I really don't know how to solve this. I need some hints to get started. I've already stated that I am thinking of solving it using AM GM HM inequality. I know nothing more than this.

Use mathematical Induction. Do you know what is it ?
 
  • #7
I think appying the binomial theorem to (1+ 1)n would also work.
 
  • #8
sankalpmittal said:
Use mathematical Induction. Do you know what is it ?

Yes I know but I'm not required to use it. Also I'm not good at it.
 
  • #9
utkarshakash said:
Yes I know but I'm not required to use it. Also I'm not good at it.

This might be your chance to improve on your technique then. Problems like this, beginning with "Prove for any n...", practically yell out "Use induction on me!" And mostly it's the easiest way to solve them.

I can see the binomial theorem being used to start 2n = (1+1)n = 1 + n Ʃ..., but getting from that sum to √2n-1 may be more work that induction would be. Maybe I'm overlooking some obvious trick though.
 
  • #10
utkarshakash said:
Yes I know but I'm not required to use it. Also I'm not good at it.

I understand what you want... But believe instead of using "Arithematic mean ≥ Geometric mean" , its more easier to use mathematical induction anyways...

But you want to use progression and series only... Ok , well , I throw you off a hint :

2n =2x2n-1.Now use A.M≥G.M here..
 

FAQ: Can AM-GM inequality prove 2^n > 1+n\sqrt{2^{n-1}} for positive integers?

What is an inequality?

An inequality is a mathematical statement that compares two quantities and shows their relative sizes. It uses symbols such as <, >, ≤, and ≥ to represent the relationship between the two quantities.

Why is proving an inequality important?

Proving an inequality is important because it allows us to show that a certain relationship between two quantities is always true. It provides a rigorous way to verify if a statement is valid or not.

What are the steps to proving an inequality?

The general steps to proving an inequality are: 1) Start with the given inequality, 2) Simplify or manipulate the inequality using algebraic properties, 3) Use logical reasoning to show that the inequality is true for all values of the variables involved, and 4) Write a conclusion stating that the inequality has been proven.

What are some common techniques used to prove inequalities?

Some common techniques used to prove inequalities include: 1) Induction, where the inequality is shown to be true for a base case and then for all other cases, 2) Contradiction, where we assume the opposite of the inequality and show that it leads to a contradiction, and 3) Rearrangement, where the given inequality is rearranged in a way that makes it easier to prove.

Are there any tips for proving inequalities?

Yes, some tips for proving inequalities include: 1) Start with the simpler side of the inequality, 2) Use properties of inequalities, such as the transitive property, to your advantage, 3) Work with absolute values when necessary, and 4) Use specific values for the variables to help guide your proof.

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