_{Disclaimer: this is a summary of section 3.2 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical,most of the equations are screenshots of the book and the same examples are treated.}

More Direct Proofs

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In creating direct proof we have to consider our hypotheses and look for a way to move from P to Q.

There are instances when we’re going to be stuck in the hypotheses without any clue on the direction towards the conclusion. The usual advice for this kind of situation is this:

“Try working backward from the conclusion.”

For our case study today, we’ll prove the “arithmetic-geometric mean inequality” – a problem that is a great example of working backward.

Arithmetic mean:

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We’re familiar with the idea of the arithmetic mean of two numbers a and b as, (a + b)/2. Consider the following sequence of numbers:

Arithmetic sequences are characterized by the property that the difference between successive terms is a constant. The blank in the sequence can be filled using the arithmetic mean of the surrounding entries, (23 + 37) / 2 = 30.

Geometric mean:

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Consider the following sequence,

Geometric sequences have the property that the ratio of successive terms is a constant. For this case, the 5th sequence can be filled by using the geometric mean of its surrounding entries, sqrt(24*96) = 48.

What we just saw is the two concept of mean. Interestingly, they can be compared, we can create a relational equivalence between the two concepts of mean. They call it the arithmetic-geometric mean inequality.

Arithmetic-Geometric Mean Inequality

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The arithmetic-geometric inequality is stated in symbolic form as,

Forwards-backwards method

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We were only given the information that a and b are non-negative real numbers – which is of no use as a starting point for our proof. For this example, we have little choice but to work backward from the conclusion.

It is shown in the table below, in the backward method column, how we started from the conclusion, or from the statement that we were trying to solve and work our way down until we reach a point of an accepted truth. (In our case, the square of any real number (e.g. (a-b) ) is greater than or equal to zero). We then rearrange the statement in the opposite order from the way they were discovered as shown in the right column of the table.

Backward Method	Forward method

One important step we should really check is the squaring on both sides of the inequality. It should be obvious to you that,

Also, note that: the square of any real number is greater than or equal to zero.

Quod Erat Demonstrandum !!

And we're done.

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Addendum

(my childhood was spent solving this kind of sequences, only now did I learn that one can actually use the neighboring numbers to solve for the missing one.)

Reference:

1. A Gentle Introduction to the Art of Mathematics by Joe Field

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