If \(\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}, \ldots \mathrm{S}_{\mathrm{n}}\) are the sums of infinite G. P.s. whose first terms are \(1,2,3, \ldots, \mathrm{n}\) and whose common ratios are \((1 / 2)\), \((1 / 3),(1 / 4), \ldots[1 /(\mathrm{n}+1)]\) respectively, then \(^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{~S}_{\mathrm{i}}=\) (A) \([\\{n(n+3)\\} / 2]\) (B) \([\\{n(n+4)\\} / 2]\) (C) \([\\{n(n-3)\\} / 2]\) (D) \([\\{n(n+1)\\} / 2]\)

When we talk about a geometric progression (GP), we refer to a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Understanding the nature of a geometric progression is crucial when dealing with series and sequences in mathematics.

For example, a GP with a first term of 1 and a common ratio of 1/2 would look like this: 1, 1/2, 1/4, 1/8, and so on. The progression can continue infinitely, getting smaller and smaller. One fascinating aspect of an infinite GP is that its sum can be finite if the absolute value of the common ratio is less than 1. This concept plays a pivotal role in solving complex problems related to series and is defined by the formula:
\[\begin{equation}S = \frac{\text{First Term}}{1 - \text{Common Ratio}}\end{equation}\]
The ability to calculate the sum of an infinite GP is foundational for understanding various phenomena in disciplines like physics, economics, and computer science, making it a powerful tool in the mathematician's arsenal.

Contrasting with the geometric progression, an arithmetic series is formed by adding the successive terms of an arithmetic sequence, where each term differs from the next by a constant amount, referred to as the common difference. This type of series is represented algebraically by:

\[\begin{equation}S = \frac{n}{2}(a_1 + a_n)\end{equation}\]
where S is the sum of the first n terms of the series, a_1 is the first term, and a_n is the nth term. In the textbook exercise, the formula helps to simplify the step where the sum of the GP sums transforms into an arithmetic series, which enables the use of this formula to find the cumulative sum. Important in countless applications, understanding the arithmetic series formula is fundamental for solving real-world problems that involve adding up sequences of numbers, such as computing interest or analyzing patterns in data.

The terms 'sequence' and 'series' are often used in the study of functions and sums in mathematics. A sequence is an ordered list of numbers that has a particular pattern. Each of the numbers in a sequence is called a term. To form a series, you sum the terms of a sequence. While a sequence is a list, a series is a single number, if it converges.

For instance, in sequences like the ones mentioned earlier, both geometric and arithmetic, each has its unique pattern. The former multiplies by a common ratio, while the latter adds a common difference. Sequences can be finite or infinite; when discussing series, particularly in the context of our textbook exercise, we're often interested in the sum of infinite sequences. This sum can be surprisingly finite and exact for infinite series with certain characteristics. The sum of a series is significant in finding patterns, solving complex mathematical problems, and modeling real-life situations. Understanding sequence and series is a fundamental part of mathematics education because it builds the groundwork for more advanced concepts.