Animated Solution for Mathematics - Sequence and Series: The sum $\frac{3\cdot 1^3}{1^2}+\frac{5\cdot (1^3+2^3)}{1^2+2^2}+\frac{7\cdot (1^3+2^3+3^3)}{1^2+2^2+3^2}+\dots$ is :

Imagine you are standing before a complex, intimidating series. It looks like a mountain of fractions, each one more daunting than the last. But in the world of JEE Advanced, every mountain has a path.

Our goal today is to find the sum of the first ten terms of the series:

The secret to conquering this is to stop looking at the whole and start looking at the DNA of a single term, $T_n$. When we isolate the $n$-th term, the chaos begins to organize itself into a beautiful, predictable pattern.

Let us break down $T_n$. The numerator has two distinct parts: the multipliers ($3,5,7,\dots$) and the sum of cubes (${\sum }_{k=1}^n{k}^3$).

The multipliers form an arithmetic progression where the $n$-th term is $(2n+1)$. The denominator is the sum of squares, ${\sum }_{k=1}^n{k}^2$.

Thus, our general term is:

This is the moment where the problem shifts from a terrifying series to a manageable algebraic expression. We are not just calculating; we are translating the language of the series into the language of algebra.

Now, we bring in our heavy artillery: the standard summation formulas. You must have these memorized as tools in your belt.

The sum of cubes is:

The sum of squares is:

When we substitute these into our expression for $T_n$, it might look messy at first:

Do not panic! This is where the magic happens. The $(2n+1)$ in the numerator and denominator will cancel out, as will one factor of $n$ and one factor of $(n+1)$.

Watch closely as the complexity evaporates. After the cancellations, we are left with:

This is the reward for your patience. A complex fraction has collapsed into a simple quadratic.

Now, finding the sum of the first ten terms is no longer a Herculean task; it is a simple summation:

We can pull the constant $\frac{3}{2}$ outside, leaving us with:

We are in the home stretch. We know that:

And:

Adding these together, we get $385+55=440$. Finally, we multiply by our constant:

There it is. The answer is 660. You didn't just solve a problem; you navigated a logical path, used your tools with precision, and found the elegant truth hidden beneath the surface.