For the following exercises, write the first eight terms of...

For the following exercises, write the first eight terms of the piecewise sequence.
$$
a_{n}=\left\{\begin{array}{ll}{(-2)^{n}-2} & {\text { if } n \text { is even }} \\ {(3)^{n-1}} & {\text { if } n \text { is odd }}\end{array}\right.
$$

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Key Concepts

Exponential Functions

Exponential functions, where a constant base is raised to a variable exponent, are fundamental in constructing sequences. When these functions are part of a sequence’s definition, they often dictate the rapid growth or decay of the terms.

Piecewise-Defined Sequences

A piecewise-defined sequence is one where the expression for generating the sequence’s terms changes based on the index. This requires evaluating different formulas for different parts of the domain, often based on conditions like the index being even or odd.

Even and Odd Indexing

The concept of even and odd indexing is used to determine which part of the piecewise definition applies to a particular term. Understanding and distinguishing these conditions is critical when the formula for the sequence depends on whether the index is even or odd.

Term Evaluation

Term evaluation in sequences involves substituting specific index values into the defined formulas. For piecewise sequences, it means carefully applying the correct formula based on the given conditions to calculate each term in the sequence.

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