2 Attachment(s)

Invariance Principle Question - from Arthur Engels book

Quoting from Arthur Engel's "Problem Solving Strategies":

Attachment 24433

To the part in **red**, how is this possible? If $\displaystyle x_{n}$ keeps increasing, $\displaystyle y_{n}$ has to decrease in order to keep the product of $\displaystyle x_{n}$ and $\displaystyle y_{n}$ constant at $\displaystyle ab$.

To the part in **blue**, I don't get this.

To the part in **green**, isn't harmonic and arithmetic mean equal in magnitude always?

Thanks. Also I don't get the inequality which he has generated. What is he trying to show by $\displaystyle x_{n+1} - y_{n+1}$?

Re: Invariance Principle Question - from Arthur Engels book

Hi, cosmonavt.

If I'm reading this right, the red part is what he's proving with the blue and green lines, so it's not supposed to make perfect sense until those other parts are established.

The blue part is how you find the midpoint between two real numbers. If x and y are real numbers, then (x+y)/2 is the midpoint between x & y.

The green part comes from an inequality that is used in analysis frequently. The inequality states that for nonnegative numbers x and y:

$\displaystyle \sqrt{xy}\leq \frac{x+y}{2}$.

The left side of the inequality is known as the "Geometric mean" and the right hand side is known as the "Arithmetic mean."

Note: There is a more general statement of the above inequality. If you're curious you can find it here Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia

Does that clear things up? Let me know. Good luck!

Re: Invariance Principle Question - from Arthur Engels book

Thanks for the reply but it doesn't clear anything. FIrstly, I know that $\displaystyle x_{n+1}$ is halfway between $\displaystyle x_{n}$ and $\displaystyle y_{n}$ but how is it showing that $\displaystyle y_{n}$ is always less than $\displaystyle x_{n}$?

Secondly, he didn't say geometric mean, he said harmonic mean. Harmonic mean and geometric mean have the same magnitude, always.

Re: Invariance Principle Question - from Arthur Engels book

The statement

Quote:

Harmonic mean and geometric mean have the same magnitude, always.

is incorrect. The relationship is actually that the harmonic mean is less than or equal to the geometric mean - see Theorem 2 of http://faculty.ccc.edu/mhidegkuti/sh...eans/means.pdf for a proof.

Second, you're right, he does quote the harmonic mean. But justifying the inequality while I read it, I used a method using the arithmetic/geometric mean inequality; I left the steps for you to see how this would work. If you would like a more explicit hint, I would suggest forming the ratio $\displaystyle \frac{y_{n+1}}{x_{n+1}}$ and seeing what you can do.