Animated Solution for Mathematics - Complex Numbers: If $x^2+x+1=0$, then the value of $(x+\frac{1}{x}{)}^4+(x^2+\frac{1}{x^2}{)}^4+(x^3+\frac{1}{x^3}{)}^4+\cdots +(x^{25}+\frac{1}{x^{25}}{)}^4$ is:

Imagine you are standing at the edge of a mathematical cliff, staring down at a massive, intimidating summation:

At first glance, it looks like a brute-force nightmare. But in the world of JEE Advanced, we never use a sledgehammer when a scalpel will do.

The equation $x^2+x+1=0$ is our scalpel. This is the signature of the complex cube roots of unity. When you see this, your mind should immediately jump to the unit circle in the complex plane.

The roots, $\omega$ and ${\omega }^2$, are not just numbers; they are vectors of length $1$, separated by $120$ degrees.

Before we touch the summation, let us simplify the base. If we take $x^2+x+1=0$ and divide every term by $x$, we get $x+1+\frac{1}{x}=0$.

This reveals the hidden truth: $x+\frac{1}{x}=-1$. This is the heartbeat of the problem.

Geometrically, adding the vectors $\omega$ and ${\omega }^2$ cancels out their imaginary components, leaving us with a real value of $-1$. This simple realization is the key that unlocks the entire structure.

Now, let us define our general term $T_k=(x^k+\frac{1}{x^k}{)}^4$. Because $x^3=1$, the powers of $x$ cycle every three steps.

This means our sequence of terms $T_k$ will also cycle. Let us calculate the first three:

For $k=1$, we have $T_1=(x+\frac{1}{x}{)}^4=(-1{)}^4=1$.

For $k=2$, we have $T_2=(x^2+\frac{1}{x^2}{)}^4=({\omega }^2+\omega {)}^4=(-1{)}^4=1$.

For $k=3$, we have $T_3=(x^3+\frac{1}{x^3}{)}^4=(1+1{)}^4=2^4=16$.

The pattern is $1,1,16$. It repeats every three terms.

We have $25$ terms in total. The pattern $1,1,16$ repeats as long as we have full cycles of three.

How many full cycles are there? $25$ divided by $3$ is $8$ with a remainder of $1$.

This means we have $8$ full cycles of $(1+1+16)$, which equals $18$ per cycle. So, $8\times 18=144$.

But wait, we have one term left over, the $25$th term. Since the pattern starts over, the $25$th term is the same as the $1$st term, which is $1$.

Adding this to our total, we get $144+1=145$. The complexity has melted away, leaving behind the pure, crystalline elegance of the answer: 145.