Animated Solution for Mathematics - Sequence and Series: If $(20{)}^{19}+2(21)(20{)}^{18}+3(21{)}^2(20{)}^{17}+\cdots +20(21{)}^{19}=k(20{)}^{19}$, then $k$ is equal to ____.

Welcome, future engineers! Today, we are going to dismantle a problem that, at first glance, looks like an absolute monster. You see a long, sprawling series:

It is easy to feel overwhelmed by the sheer size of these numbers. But in the world of JEE Advanced, intimidation is just a mask for a beautiful, underlying simplicity. Let us peel back that mask together.

The first thing we must do is simplify our perspective. Look at the powers: the powers of $20$ are marching downwards from $19$ to $0$, while the powers of $21$ are climbing upwards from $0$ to $19$.

This is a classic signature of an Arithmetico-Geometric Progression (A.G.P.). To make this structure obvious, we divide the entire equation by $(20{)}^{19}$.

Suddenly, the chaos settles. If we define $r=\frac{21}{20}$, our equation transforms into the elegant form:

Now, we are speaking the language of mathematics.

How do we handle an A.G.P.? We use the 'Shift and Subtract' method. It is a foolproof, elegant strategy.

We multiply our entire series by the common ratio $r$:

Now, we align this new equation underneath our original one. By shifting the terms, we create a vertical alignment where we can subtract the two equations:

Look at what happens! The arithmetic coefficients vanish, leaving us with a clean geometric series:

The first part of our expression is a standard geometric progression with $20$ terms. We use the sum formula $S=\frac{a(r^n-1)}{r-1}$.

Substituting $a=1$ and $n=20$, we get:

Now, we bring back our value $r=\frac{21}{20}$. Notice that $1-r=-\frac{1}{20}$ and $r-1=\frac{1}{20}$.

Substituting these values, the equation becomes:

Multiplying through by $20$, we find:

Expanding this, we see the magic:

The $20r^{20}$ terms cancel out completely, leaving us with $-\frac{k}{20}=-20$. With a simple multiplication, we find:

You see? What started as a terrifying series ended in a clean, perfect integer. Keep this logic in your toolkit, and no series will ever scare you again!