The image shows a mathematical expression that is a sum of several terms, starting with:

$1 + \left(\frac{1}{2} + \frac{2}{2}\right) + \left(\frac{1}{3} + \frac{2}{3} + \frac{3}{3}\right) + \cdots + \left(\frac{1}{100} + \frac{2}{100} + \cdots + \frac{99}{100}\right)$

This appears to follow a pattern where each term of the sum is the sum of fractions where the numerators go from $1$ to one less than the denominator (which is increasing from 2 to 100).

Let's break it down:

1. First term: $1$
2. Second term: $\frac{1}{2} + \frac{2}{2} = \frac{3}{2}$
3. Third term: $\frac{1}{3} + \frac{2}{3} + \frac{3}{3} = \frac{6}{3} = 2$
4. Fourth term: $\frac{1}{4} + \frac{2}{4} + \frac{3}{4} + \frac{4}{4} = \frac{10}{4} = 2.5$
5. The final term involves the sum of fractions for denominator $100$, which would be: $\frac{1}{100} + \frac{2}{100} + \cdots + \frac{99}{100} = \frac{99 \times 100}{200} = 49.5$

This sum follows a pattern where for each $n$-th term, you are summing up $\frac{1}{n} + \frac{2}{n} + \dots + \frac{n}{n}$, which simplifies to $\frac{n(n+1)}{2n} = \frac{n+1}{2}$.

Therefore, the entire sum is:

$1 + \frac{3}{2} + 2 + 2.5 + \cdots + \frac{100}{2}$

This is an arithmetic progression from $1$ to $50$. The sum of an arithmetic progression is given by:

$S = \frac{n}{2} \left( a_1 + a_n \right)$

where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. Here, $a_1 = 1$ and $a_n = 50$, with 100 terms in total. So:

$S = \frac{100}{2} \left( 1 + 50 \right) = 50 \times 51 = 2550$

Thus, the total sum is $2550$.

Would you like to go deeper into any part of this explanation?

Here are some related questions you might find interesting:

1. What is an arithmetic progression, and how is it derived?
2. Can we generalize this sum for any series of this form?
3. How do you compute sums of fractions efficiently?
4. What are the properties of series that grow with both numerator and denominator?
5. How do partial sums change the outcome of series?

Tip: Arithmetic progression sums are very useful when dealing with regularly increasing sequences. Knowing the formula can help simplify complex sums quickly.