Animated Solution for Mathematics - Sequence and Series: Let $a,b$ be positive real numbers. If $a,A_1,A_2,b$ are in arithmetic progression, $a,G_1,G_2,b$ are in geometric progression and $a,H_1,H_2,b$ are in harmonic progression, show that $\frac{G_1{G}_2}{H_1{H}_2}=\frac{A_1+A_2}{H_1+H_2}=\frac{(2a+b)(a+2b)}{9ab}$.

Imagine you are standing on a vast, perfectly flat plain, and before you lie two markers: $a$ and $b$. These are your anchors, your positive real numbers.

Between them, we are going to weave three different tapestries: an Arithmetic Progression, a Geometric Progression, and a Harmonic Progression. This problem is not just about crunching numbers; it is about understanding the hidden symmetry that binds these sequences together.

We start with the Arithmetic Progression: $a,A_1,A_2,b$. The beauty of an arithmetic progression lies in its linearity.

When we insert $n$ arithmetic means between two numbers, the sum of those means is simply $n$ times the average of the two extremes. Here, with $n=2$, the sum $A_1+A_2$ becomes:

Next, we turn to the Geometric Progression: $a,G_1,G_2,b$. Geometric progressions are about ratios, not differences.

The product of $n$ geometric means inserted between two numbers is the square root of their product, raised to the power of $n$. For our two means, $G_1{G}_2$ is simply $(ab{)}^{\frac{2}{2}}$, which is just $ab$. We have our first two pieces of the puzzle: $A_1+A_2=a+b$ and $G_1{G}_2=ab$.

Now, we face the Harmonic Progression: $a,H_1,H_2,b$. Harmonic progressions are notoriously slippery because they do not behave linearly.

But we have a secret weapon: the reciprocal. By definition, if $a,H_1,H_2,b$ are in HP, then their reciprocals $\frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$ must form an Arithmetic Progression.

We are no longer in the land of harmonic complexity; we are back in the familiar territory of arithmetic simplicity. We know that $\frac{1}{H_1}$ and $\frac{1}{H_2}$ are two arithmetic means between $\frac{1}{a}$ and $\frac{1}{b}$. Therefore, their sum must be:

If we manipulate this, we get $\frac{H_1+H_2}{H_1{H}_2}=\frac{a+b}{ab}$. This is a pivotal moment where we have successfully linked the sum and the product of our harmonic means.

We are now ready to assemble the final proof. We need to evaluate the ratio $\frac{A_1+A_2}{H_1+H_2}$.

Substituting our known values, we get:

Now, look at the second ratio we need to prove: $\frac{G_1{G}_2}{H_1{H}_2}$. Since $G_1{G}_2=ab$, this ratio is also $\frac{ab}{H_1{H}_2}$. They are identical!

To finish the proof, we find the explicit value of $H_1{H}_2$. We return to our reciprocal AP and find the common difference $D$:

With $D$ in hand, we find the individual means:

Multiplying these, we get:

Finally, substituting this back into our ratio $\frac{ab}{H_1{H}_2}$, we obtain the final result: