Animated Solution for Mathematics - Sequence and Series: Let $A_1$ and $A_2$ be two arithmetic means and $G_1,G_2$ and $G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$ is equal to

Imagine you are standing on a number line, holding two distinct positive numbers, $a$ and $b$. These are our anchors, our extremes.

Between them, we are tasked with inserting two arithmetic means, $A_1$ and $A_2$, and three geometric means, $G_1,G_2$, and $G_3$. This problem is not just about calculation; it is about uncovering the hidden symmetry between arithmetic and geometric progressions.

First, let us look at the arithmetic side. When we insert $A_1$ and $A_2$ between $a$ and $b$, we create an Arithmetic Progression: $a,A_1,A_2,b$.

The beauty of an A.P. lies in its linearity. The sum of the means is always equal to the sum of the extremes.

Thus, we immediately find that:

Keep this relationship safe in your mind; it is the key to our final destination.

Now, let us pivot to the geometric side. We have $G_1,G_2,G_3$ inserted between $a$ and $b$, forming a Geometric Progression: $a,G_1,G_2,G_3,b$.

Let $r$ be the common ratio. Since there are five terms in total, the fifth term is $b=a{r}^4$, which gives us:

Now, we express our means: $G_1=ar$, $G_2=a{r}^2$, and $G_3=a{r}^3$. The expression we need to evaluate is $E=G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$.

Let us calculate these powers:

Finally, the term $G_1^2{G}_3^2$ is simply $(G_1{G}_3{)}^2$. Since $G_1{G}_3=(ar)(a{r}^3)=a^2{r}^4=ab$, we find that:

We are now ready to assemble our masterpiece. Substituting these values into our expression $E$, we get:

Combining the like terms, we arrive at:

Look closely at this expression. If we factor out $ab$, we are left with $ab(a^2+2ab+b^2)$. The term inside the parenthesis is a perfect square, $(a+b{)}^2$.

So, the expression simplifies to:

Now, recall our earlier findings: $a+b=A_1+A_2$ and $ab=G_1{G}_3$. Substituting these back, we get:

This is the elegance of mathematics—a complex-looking expression collapsing into a simple, beautiful identity. You have successfully navigated the relationship between means and extremes.