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Sigma notation, often represented by the Greek letter \( \Sigma \), is a compact way to represent the sum of a series of terms. This method can be quite useful since it helps to succinctly express long sums without having to write out every single term. Each term in the series is represented as a formula involving an index, which typically follows the sigma symbol. An example from our textbook includes \( \sum_{k=2}^{5}(k+1)^{2}(k-3) \), where \( k \) is the index that starts at 2 and ends at 5.

When following sigma notation, the index \( k \) takes on each sequential value in the specified range, and we calculate the term for each value of the index. In the case of our example, the range is from 2 to 5. This means we calculate \( (k+1)^{2}(k-3) \) first for \( k=2 \) and then incrementally upwards to \( k=5 \) for each subsequent term. To complete the summation, we would then add up all of these calculated values.

Series and sequences are foundational concepts in arithmetic and algebra, which deal with ordered lists of numbers. A sequence is just an ordered list, while a series is the sum of a sequence of terms. In a sequence, the terms follow a specific rule that tells you the pattern of the sequence. For instance, \( k+1 \), where \( k \) ranges from 2 to 5, represents a sequence of numbers that are derived by adding 1 to each \( k \) in the range.

In the given exercise, the series we are evaluating is created by taking the sequence of terms formed by the expression \( (k+1)^{2}(k-3) \) for each value of \( k \) from 2 to 5 and adding them together. Mathematically, this is represented as a sum because a series can be thought of as a sum of its terms. Understanding the distinction and the relation between sequences (the individual elements) and series (the sum of these elements) can greatly simplify the process of working with sigma notation.

Arithmetic operations, such as addition, subtraction, multiplication, and division, are basic mathematical procedures used in virtually all levels of mathematical discourse. In the context of our textbook example, these operations help us evaluate the terms of the series.

First, each term \( (k+1)^{2}(k-3) \) is calculated using the exponentiation operation to square the \( k+1 \) term and the multiplication operation to multiply the squared result with \( k-3 \) term. Next, to find the sum of the series, we use the addition operation, incorporating any negative terms from the subtraction operation involved in each respective term's evaluation. Specifically, in our example, once we evaluate each term for \( k=2, k=3, k=4 \) and \( k=5 \) using these arithmetic operations, we simply add the results together to get the final sum of the series.