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?
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.
utkarshakash said:Ok so assume that it is 'greater that or equal to' instead of just 'is greater than' and prove it
Ray Vickson said:No: YOU prove it, or at least show some effort towards the solution. Read the Forum rules!
RGV
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.
sankalpmittal said:Use mathematical Induction. Do you know what is it ?
utkarshakash said:Yes I know but I'm not required to use it. Also I'm not good at it.
utkarshakash said:Yes I know but I'm not required to use it. Also I'm not good at it.
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.
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.
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.
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.
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.