# Show that the inequality is true | Geometric Mean

• michonamona
In summary, the inequality (1+R_{G})^{n} \leq V is true for strictly positive numbers r_{1}, r_{2}, ... , r_{n}. This can be shown by using the arithmetic mean-geometric mean inequality, as well as taking the log of both sides and expanding the term. Another approach is to use log(1+e^x) and associate r with e^x, as suggested by the professor. Alternatively, the terms of the right side can be seen as the elementary symmetric functions of the r_j, providing another way to prove the inequality.
michonamona

## Homework Statement

Let $$r_{1}, r_{2}, ... , r_{n}$$ be strictly positive numbers. Show that the inequality

$$(1+R_{G})^{n} \leq V$$

is true. Where $$R_{G} = (r_{1}r_{2}...r_{n})^{1/n}$$ and $$V= \Pi_{k=1}^{n} (1+r_{k})$$

## The Attempt at a Solution

I've tried taking the log of both sides, as well as expanding out the term. Any insight?

Thanks,
M

Use the arithmetic mean-geometric mean inequality... several times.

Any other insights?

The prof hinted that we should use log(1+e^x) and associate r with e^x.

That's an entirely different way to approach it. The approach I was thinking of uses the fact that the terms of the right side are the elementary symmetric functions of the $$r_j$$.

## 1. What is the definition of geometric mean?

The geometric mean is a type of average that is calculated by taking the nth root of the product of n numbers. For example, the geometric mean of 2 and 8 is √(2*8) = 4, since 4 is the number that, when multiplied by itself, gives 2 and 8.

## 2. How is the geometric mean different from other types of averages?

The geometric mean differs from other types of averages, such as arithmetic mean and median, because it takes into account the relative magnitudes of the numbers. This makes it useful for calculating average growth rates, ratios, and other quantities that involve multiplication.

## 3. How do you show that an inequality is true using geometric mean?

To show that an inequality is true using geometric mean, you need to prove that the nth root of the product of n numbers is less than or equal to the arithmetic mean of the n numbers. This can be done by using the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean of the same set of numbers.

## 4. Can the geometric mean be used for any set of numbers?

Yes, the geometric mean can be used for any set of positive numbers. However, it is not defined for negative numbers or 0, as you cannot take the nth root of a negative number or 0.

## 5. How is the geometric mean used in real-life applications?

The geometric mean has many real-life applications, such as in finance to calculate average return rates, in biology to measure growth rates, and in statistics to compare data sets. It is also used in various industries, including agriculture, engineering, and physics, to calculate average values of quantities involving multiplication.

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