Animated Solution for Mathematics - Sequence and Series: The rational number, which equals the number $2.357$ with recurring decimal is

Welcome, fellow traveler on the road to JEE excellence. Today, we are going to demystify a concept that often feels like a simple arithmetic trick but is, in reality, a beautiful gateway into the world of infinite series.

We are looking at the number $2.\overline{357}$. At first glance, it looks like a simple decimal, but that bar over the $357$ is a signal—a signal that we are dealing with an infinite process captured in a finite symbol.

When we see $2.\overline{357}$, we must learn to see it not as a static value, but as a sum. We can peel away the integer part, leaving us with $2+0.357357357\dots$.

This is our first act of mathematical liberation. By separating the integer $2$, we isolate the complexity of the repeating decimal, allowing us to focus our analytical lens on the infinite tail.

Now, let us look at $0.357357357\dots$ with the eyes of a mathematician. We can write this as a sum of fractions based on their place value:

Do you see the pattern? Each term is exactly $\frac{1}{1000}$ of the term before it. This is the heartbeat of a Geometric Progression.

We have a first term $a=\frac{357}{1000}$ and a common ratio $r=\frac{1}{1000}$. Because the magnitude of our ratio $|r|<1$, this infinite sum converges to a finite, elegant value. We invoke the sacred formula for the sum of an infinite G.P.:

Substituting our values into this formula feels like watching gears click into place. We have:

Look at the denominator: $1-\frac{1}{1000}$ becomes $\frac{999}{1000}$. When we divide the fractions, the $1000$ in the numerator and the $1000$ in the denominator perform a perfect, silent cancellation.

We are left with $S=\frac{357}{999}$. There is a profound satisfaction in seeing such a complex infinite process collapse into such a clean, rational fraction.

We are almost home. We must now reunite our integer $2$ with the fraction we just derived. We add them together:

To combine these, we find a common denominator, which is $999$. This gives us:

Calculating the numerator, $2\times 999$ gives us $1998$. Adding $357$ to $1998$ yields $2355$.

Thus, our final rational number is $\frac{2355}{999}$.

As you look at this result, remember that you didn't just solve a problem; you translated an infinite, repeating reality into a precise, finite ratio. This is the essence of physics and mathematics—finding the order within the infinite.

Keep this curiosity alive, keep questioning the 'why' behind every step, and you will find that even the most daunting problems are just stories waiting to be told. You have the tools, you have the logic, and now, you have the perspective. Go forth and conquer the next challenge!