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In mathematics, sequence notation is a method used to represent the elements in a sequence, a list of numbers arranged in a particular order. A sequence is denoted by an alphabetical letter with subscripted integers. For example, the expression (a_n = (-3)^n) is a notation for a sequence where n represents the position of the term within the sequence, and a_n represents the nth term itself.

When generating a sequence, you'll frequently replace the n with consecutive integers to find the specific terms. In the case of our exercise, replacing n with 1, 2, 3, and 4 yields the first four terms of the geometric sequence given by the formula a_n = (-3)^n. This approach makes determining the pattern of any sequence straightforward and calculating specific terms even easier.

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When a number (known as the base) is raised to the power of another number (the exponent), it signifies multiple multiplication of the base. In our exercise, the sequence involves exponentiating the base -3.

For example, (-3)^1 is simply -3, while (-3)^2 is -3 multiplied by -3, which equals 9. It’s essential to highlight that when the exponent is even, the result will always be positive since a negative number times itself an even number of times results in a positive product. However, with an odd exponent, the result will remain negative. Understanding this concept is crucial for dealing with sequences involving alternating signs like the one in our problem.

The fundamental difference between a sequence and a series in mathematics must be made clear. A sequence, as seen with a_n = (-3)^n, is a list of numbers displaying a specific order based on a rule or formula. Each number in the sequence is called a term, and in the context of our exercise, we have calculated the first four terms. A series, on the other hand, refers to the sum of the terms in a sequence.

If we were to convert our geometric sequence into a series, we would add the terms together. It’s important to be familiar with the behavior of sequences to understand how their corresponding series will behave—for instance, whether the sum of the series would converge to a limit or not, especially when dealing with an infinite number of terms.