Animated Solution for Mathematics - Sequence and Series: Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

Imagine you are standing before a complex algebraic puzzle. You are given a sum, $a+b+c+d=11$, and you are asked to maximize a product, $P=a^5{b}^3{c}^{2}d$.

At first glance, this looks like a nightmare of multivariable calculus. But in the world of JEE Advanced, whenever you see a sum of positive real numbers and a request to maximize their product, there is one tool that stands above all others: the Arithmetic Mean-Geometric Mean (AM-GM) Inequality.

A Masterclass in Manipulation

The AM-GM inequality states that for any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean. Mathematically, this is expressed as:

But here is the trap: if you apply this directly to $a,b,c,$ and $d$, you get the product $abcd$. That is not what we want; we need $a^5{b}^3{c}^{2}d$.

This is where the Splitting Strategy comes into play. To get $a^5$, we must break $a$ into five equal pieces, each of size $\frac{a}{5}$. When we multiply these five pieces together, we get $(\frac{a}{5}{)}^5$, which perfectly generates the $a^5$ term we desire.

We apply this same logic to $b$ (splitting it into three parts of $\frac{b}{3}$) and $c$ (splitting it into two parts of $\frac{c}{2}$). Since $d$ has an exponent of $1$, we leave it as a single term.

Now, let us count our terms. We have $5$ terms of $\frac{a}{5}$, $3$ terms of $\frac{b}{3}$, $2$ terms of $\frac{c}{2}$, and $1$ term of $d$. The total number of terms is $n=5+3+2+1=11$.

Let us calculate the Arithmetic Mean ($AM$):

Notice the magic? The numerator simplifies beautifully to $a+b+c+d$. Since we know this sum is $11$, our $AM=\frac{11}{11}=1$.

Now, let us look at the Geometric Mean ($GM$):

Applying the inequality $AM\ge GM$, we get:

Raising both sides to the power of $11$, we find:

This means our maximum product is $P_{max}=5^5\cdot 3^3\cdot 2^2$. Calculating this, we get:

The problem states $P_{max}=3750\beta$. Equating $3750\beta =337500$, we find $\beta =90$. You have just conquered a classic optimization problem using the elegance of pure algebra.