Intereting Posts

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Triangular Factorials
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Weak convergence and strong convergence
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Nested Radicals: $\sqrt{a+\sqrt{2a+\sqrt{3a+\ldots}}}$
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Is there a partial sum formula for the Harmonic Series?
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Some questions about mathematical content of Gödel's Completeness Theorem

The arithmetic mean of $k$ numbers $a_1, a_2, \ldots, a_k$ is their average $\frac{a_1+a_2+\cdots+a_k}{k}=AM$. Their geometric mean

is $\sqrt[k]{a_1a_2\cdots a_k}=GM$. I am asked to show this:

Use induction to prove: If $k=2^n$ and if all the numbers $a_1, a_2, \ldots, a_k$ are nonnegative, then $AM \geq GM$.

I’ll be honest, I have no work for this problem. I’ve looked into many examples of strong induction, but most of them are abstract. Please give me insight for what method to best utilize for strong induction.

- What exactly is the difference between weak and strong induction?
- PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
- Prove by induction that for all $n$, $8$ is a factor of $7^{2n+1} +1$
- Proving by strong induction for a sequence of integers, $2^n$ divides term $n$
- Fibonacci using proof by induction: $\sum_{i=1}^{n-2}F_i=F_n-2$
- How does one actually show from associativity that one can drop parentheses?

- Geometric mean never exceeds arithmetic mean
- What are some examples of induction where the base case is difficult but the inductive step is trivial?
- Revisited: Binomial Theorem: An Inductive Proof
- Is it possible to play the Tower of Hanoi with fewer than $2^n-1$ moves?
- Examples where it is easier to prove more than less
- There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$
- show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
- Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
- Trying to generalize an inequality from Jitsuro Nagura: Does this work?
- When does the equality hold in the triangle inequality?

The case $k=2^n$ can be proved using ordinary induction on $n$. The induction argument is structurally natural, and there is no reason to think that there is a smoother proof by strong induction.

The standard proof, due to Cauchy, can be found here. Full details are given, so the link should be sufficient. If you have trouble with the induction step, work out what the proof says for the particular cases $k=4$, then $k=8$, and everything will be clear.

Cauchy actually proved the full Arithmetic Mean/Geometric Mean Inequality, by first dealing with the cases $k=2^n$, and then going backwards to deal with $k$ not a power of $2$. It is an elementary but very clever proof.

The use of induction (first ordinary induction to prove the result for powers of $2$, then upward-downward induction for the general case) to prove the Arithmetic-Geometric Mean Inequality appears in my lecture notes on induction (Theorem 21 in $\S 12$).

I agree that this argument is somewhat tricky. I am currently teaching a “Spivak calculus course” (i.e., very strong, very committed university freshmen) and I put this argument into my notes (which originated in a different undergraduate course) expressly for the current course but have not, so far, had a chance to mention it in class. I think many if not most young students would have trouble supplying a complete proof.

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