Animated Solution for Mathematics - Sequence and Series: The sum ${\sum }_{k=1}^{20}(1+2+3+....+k)$ is _____________.

Imagine you are standing before a grand staircase, but instead of simple steps, each level is itself a smaller staircase. This is the essence of our problem: a nested summation. We are asked to find the sum ${\sum }_{k=1}^{20}(1+2+3+\cdots +k)$.

At first glance, it might look like a daunting task, but let us break it down with the precision of a master architect.

Look closely at the inner term: $1+2+3+\cdots +k$. This is not just a random sequence; it is the sum of the first $k$ natural numbers, which forms a triangular number.

For $k=1$, we have $1$. For $k=2$, we have $1+2=3$. For $k=3$, we have $1+2+3=6$. We are essentially stacking these triangular numbers twenty times.

To handle this, we use the standard formula for the sum of the first $n$ natural numbers: ${\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}$. By applying this to our inner term, we collapse the entire inner series into a single, elegant expression:

Now, our problem transforms. We are no longer dealing with a nested series, but a single, manageable summation:

This is the power of algebra—it turns complexity into clarity. We can use the linearity property of summations to pull the constant $\frac{1}{2}$ outside the summation sign, leaving us with:

We have successfully reduced a double summation into two standard, well-known series.

To finish this, we need our JEE toolkit. We know the formulas for the sum of the first $n$ integers and the sum of the first $n$ squares:

With $n=20$, we substitute these values. For the sum of squares, we get:

For the sum of integers, we get:

We are almost there. Adding these two results gives us $2870+210=3080$.

Finally, we multiply by the constant $\frac{1}{2}$ we pulled out earlier:

The journey from a complex nested series to a single integer is complete. Remember, in physics and mathematics, the secret is not to fear the complexity, but to systematically dismantle it until only the truth remains. The final answer is 1540.