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Understanding an arithmetic series is crucial for working with sequences of numbers where the difference between consecutive terms is constant. This condition characterizes arithmetic series and is fundamental in various mathematical calculations. For example, the sum of the first ‘n’ odd numbers can be expressed as the summation \(\sum_{i=1}^{n}(2i - 1)\). In simpler terms, an arithmetic series can be visualized as a lineup of evenly spaced points on a number line.

To find the sum of an arithmetic series, you don’t need to add up all terms individually. A formula exists to simplify the process: \(S_n = \frac{n (a_1 + a_n)}{2}\), where \(S_n\) is the sum of the first ‘n’ terms, \(a_1\) is the first term, and \(a_n\) is the last term. Applying this to our example series, the sum can be calculated swiftly without enumerating each term, showcasing the power of mathematical shortcuts for efficient problem-solving.

A closed-form expression is a mathematical treasure. It converts a potentially infinite process, like adding up a series, into a single, elegant formula. Imagine the laborious task of summing each term of a long arithmetic series – it's akin to counting grains of sand! Closed-form expressions, however, offer an umbrella shield against this mathematical downpour.

As illustrated in the exercise, the summation \(\sum_{i=1}^{n}(2i - 1)\) can be condensed into a neat closed-form expression: first by recognizing it as the combination of two series and then by applying known formulas for arithmetic and constant series. This approach dramatically simplifies the task, showcasing the might of closed-form expressions in converting extensive summations into manageable formulas.

Big Omega notation, expressed as \(\Omega\), serves as a mathematical sentinel, standing guard to signify the lower bound of an algorithm's running time or, in our case, the growth of functions as the input expands without limit. It's like the safety net under a dauntingly high arithmetic act, ensuring that the function's growth will not fall below a certain threshold.

In the verification of \(\sum_{i=1}^{n}(2i - 1) = \Omega(n^2)\), we effectively place a baseline of performance, stating that the growth rate of our summation is at least as rapid as \(n^2\), if not faster. Through the reasoning in our solution, comparing \(n(n + 1) - n\) to \(n^2\), we assure ourselves of this minimum growth rate, thus reinforcing the notation's role as a foundational concept in algorithmic analysis and complexity.

Discrete mathematics is a branch of mathematics that revels in the study of distinct and separable mathematical structures, as opposed to the continuous. Imagine a constellation of stars as opposed to the smooth slide of a comet's tail — that's discrete versus continuous. This domain includes topics like sets, graphs, logic, and, significantly for our discussion, series and sequences.

Summation notation and the investigation of series, as we've traversed in this exercise, are grounded in the principles of discrete mathematics. Recognizing the discrete nature of our series allows us to apply specific techniques and formulas tailored for these kinds of mathematical entities. Discrete mathematics thus becomes the toolbox from which we extract the instruments needed to unlock and simplify complex problems like the one we've tackled here.