Animated Solution for Mathematics - Sequence and Series: The harmonic mean of two numbers is $4$. Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2A+G^2=27$. Find the two numbers.

A JEE Algebra Journey

Welcome, future engineer! Today, we are going to unravel a problem that sits at the very heart of algebraic beauty. In the JEE Advanced curriculum, you will often encounter problems that seem to be about finding two mysterious numbers, but are actually about understanding the deep, interconnected relationships between different types of averages: the Arithmetic Mean ($A$), the Geometric Mean ($G$), and the Harmonic Mean ($H$).

Decoding the Definitions

Imagine you are standing before a puzzle. You have two numbers, $a$ and $b$. We are given that their harmonic mean $H$ is $4$. We are also given a specific, seemingly complex relation: $2A+G^2=27$.

Our mission is to find $a$ and $b$. To do this, we must first translate these words into the language of mathematics. Recall the definitions that form our toolkit:

1. The Arithmetic Mean: $A=\frac{a+b}{2}$ 2. The Geometric Mean: $G=\sqrt{ab}$, which implies $G^2=ab$ 3. The Harmonic Mean: $H=\frac{2ab}{a+b}$

We start with the harmonic mean. We are told $H=4$. Substituting the formula, we get $4=\frac{2ab}{a+b}$.

With a little algebraic manipulation, multiplying both sides by $(a+b)$ and dividing by $2$, we find a beautiful, simple bridge: $ab=2(a+b)$. Let us hold onto this as our Equation (1). This is the key that will unlock the rest of the problem.

The Algebraic Bridge

Now, let us look at the second piece of information: $2A+G^2=27$. This looks intimidating, but let us substitute our definitions into it. We know $A=\frac{a+b}{2}$ and $G^2=ab$.

Substituting these, the equation becomes:

Look at that! The $2$ in the numerator and the $2$ in the denominator cancel out perfectly. We are left with $(a+b)+ab=27$.

This is where the magic happens. We have two variables, but we have a relationship between them from Equation (1). We can replace $ab$ with $2(a+b)$.

Substituting this into our simplified equation, we get:

This simplifies to $3(a+b)=27$. Dividing by $3$, we find that the sum of our two numbers is $a+b=9$. Now, using Equation (1), we can easily find the product: $ab=2(9)=18$.

The Quadratic Revelation

We now know the sum ($S=9$) and the product ($P=18$) of our two numbers. In the world of algebra, if you know the sum and product of two roots, you can construct the quadratic equation they satisfy.

The equation is given by $x^2-Sx+P=0$. Substituting our values, we get:

This is a classic quadratic equation. We need two numbers that multiply to $18$ and add to $-9$. Factoring this, we get $(x-6)(x-3)=0$.

The roots are $x=6$ and $x=3$. Thus, our two numbers are $3$ and $6$.

Isn't it elegant? We started with a complex-looking relation and, by systematically using the definitions of $A,G,$ and $H$, we reduced it to a simple quadratic equation.

This is the essence of JEE preparation: not just memorizing formulas, but understanding how to manipulate them to reveal the underlying structure of the problem. Keep practicing, stay curious, and remember that every complex problem is just a collection of simple, beautiful steps waiting to be discovered.