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A sequence in mathematics is an ordered list of numbers that follows a particular pattern or rule. In the context of arithmetic sequences, the rule consists of adding (or subtracting) a constant value to the previous term to obtain the next term.

In the exercise provided, the arithmetic sequence is defined by the algebraic expression \(t_{n}=2n-1, n \geq 1\). It means that to find any term in the sequence, we multiply the term's position (\(n\)) by 2 and then subtract 1. For example, to get the first term (\(t_1\)), we substitute \(n\) with 1, resulting in \(2(1)-1 = 1\). Similarly, the second term (\(t_2\)) is \(2(2)-1 = 3\), and so on.

Understanding the pattern in a sequence allows you to predict future terms and also calculate the sum of a certain number of terms.

Summation notation, represented by the Greek letter sigma (\(\Sigma\)), is a concise way to write the sum of a series of terms. The notation includes an expression for the terms to be added, the index of summation (usually \(i\) or \(n\)), the starting value, and the ending value of the index.

For instance, the sum \(\sum_{i=1}^{3} t_{i}\) in the given exercise indicates that we want to add up the first three terms of the sequence \(t\). The \(i\) is the index of summation that starts from 1 and goes up to 3. Summation notation is incredibly useful for encapsulating the addition of terms in a more manageable form, especially when dealing with large sequences or more complex series.

To calculate the sum stated in the exercise, we find the first three terms of the sequence and add them together in accordance with the summation notation, resulting in the total sum of 9.

Algebraic expressions are combinations of numbers, variables (letters representing unknown or variable quantities), and arithmetic operations like addition, subtraction, multiplication, and division. In sequences, algebraic expressions define the rule for generating terms.

In our exercise, \(t_{n}=2n-1\) is the algebraic expression for finding the \(n\)-th term of the sequence. The \(n\) serves as the variable that can take on any positive value, and the expression outlines the steps to calculate the corresponding term.

Learning to work with algebraic expressions is crucial for understanding sequences and the broader landscape of algebra. It allows you to create general solutions that can apply to a range of problems instead of relying on a case-by-case basis.