Sequences and Series – Geometric Mean Never Exceeds Arithmetic Mean

algebra-precalculusinductioninequalitysequences-and-seriesstatistics

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS – LHS > 0, splitting the fraction, fractional exponents, etc.)

The geometric mean of $n$ positive numbers $a_1, a_2,\ldots,a_n$ is $\sqrt[n]{a_1a_2 \ldots a_n}$ and their arithmetic mean is $\frac{a_1+a_2+\ldots+a_n}{n}$. If $a_1, a_2,\ldots,a_n$ are $n$ positive real numbers, prove by induction that their geometric mean is always smaller than or equal to their arithmetic mean, i.e. $\sqrt[n]{a_1a_2\ldots a_n} \leq \frac{a_1+a_2+\ldots+a_n}{n}$

Best Answer

Hint:

Using induction:

  1. Show the base case $n = 2$.
  2. Show that if the statement is true for $2^n$, then it is true for $2^{n+1}$.
  3. Show that if the statement is true for $n$, then it is true for $n - 1$.

(This method of induction is sometimes called Cauchy induction).

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