First Five Terms of the Sequence for an

Given the sequence $a_n = (-1)^n (n - 2)$, find the first five terms of the sequence.

Sequences, a fundamental topic in mathematics, involve the ordered arrangement of numbers and often lay the groundwork for understanding series and convergence in calculus. The sequence in this problem, $(-1)^n (n - 2)$, alternates in sign due to the term $(-1)^n$. This makes it an alternating sequence, where the sign changes with each subsequent term depending on whether $n$ is odd or even. Understanding alternating sequences is crucial as they appear frequently, especially when dealing with complex series problems.

A key strategy in solving sequence problems is to identify the general formula, which here involves both an arithmetic component $(n - 2)$ and an alternating sign component $(-1)^n$. To find specific terms of the sequence, one should substitute the desired values of $n$ into the formula. This process requires attention to detail, especially in keeping track of positive and negative signs due to the alternating factor.

The concepts explored through this sequence problem include understanding alternating sequences, evaluating terms by substituting index values, and analyzing the behavior of sequences as $n$ varies. These skills lay a foundational understanding necessary for tackling more complex topics in series and calculus, such as convergence and divergence of series.

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