Animated Solution for Mathematics - Sequence and Series: The product of $n$ positive numbers is unity Then their sum is

A Journey Through AM-GM

Imagine you are standing before a set of $n$ positive numbers, $x_1,x_2,\dots ,x_n$. You are told that their product is locked in a perfect balance: $x_1\cdot x_2\cdot \cdots \cdot x_n=1$.

Your task is to uncover the secret of their sum. At first glance, this might seem like a vague request, but mathematics loves to hide rigid structures behind simple conditions.

Let us begin our exploration by simplifying the world.

Suppose we only have two numbers, $x_1$ and $x_2$. The condition is $x_1\cdot x_2=1$, which implies $x_2=\frac{1}{x_1}$.

The sum is $S=x_1+\frac{1}{x_1}$. If you were to graph this function, you would see a beautiful curve that dips down and then rises sharply.

The minimum occurs exactly when $x_1=1$, yielding a sum of $1+1=2$. For any other positive value, the sum is strictly greater than $2$. This is our first clue: the sum is bounded from below.

Now, let us scale this up to $n$ numbers. We need a tool that bridges the gap between the sum and the product.

Enter the Arithmetic Mean-Geometric Mean (AM-GM) Inequality. It is one of the most elegant theorems in algebra, stating that for any set of positive numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean:

This inequality is the key that unlocks the problem. It tells us that the average of our numbers is constrained by their product.

Let us apply this to our specific case. We know that the product $x_1\cdot x_2\cdot \cdots \cdot x_n=1$.

Substituting this into the inequality, the right side becomes $\sqrt[n]{1}$, which is simply $1$. The inequality transforms into:

Now, with a simple algebraic step, we multiply both sides by $n$:

This is the moment of clarity. The sum of these $n$ numbers is never less than $n$.

It can be $n$ (when all numbers are $1$), or it can be larger, but it can never drop below that threshold. We have successfully navigated the complexity of $n$ variables by relying on the fundamental symmetry of the AM-GM inequality.

Remember, in JEE Advanced, the most complex-looking problems often yield to the most elegant principles. Keep this tool in your arsenal, and you will find that even the most daunting algebraic expressions have a simple, beautiful core. The final result is ${\sum }_{i=1}^n{x}_i\ge n$.