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A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The geometric sequence formula for the nth term (\(a_n\)) is expressed as:
\[a_n = a_1 \times r^{(n-1)}\]
Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Using this formula, you can find any term of a geometric sequence, given that you know the first term and the common ratio. It's a powerful tool for both identifying patterns within a sequence and predicting future terms.

Calculating the terms of a geometric sequence involves applying the geometric sequence formula. When computing terms beyond the first, it's crucial to carefully follow each step to ensure accuracy. As seen in the provided exercise, each subsequent term is derived by multiplying the preceding term by the common ratio raised to the power of the term's position minus one. For instance, to calculate the second term, you raise the common ratio to the first power:
\[a_{2}= a_{1} \times r^{1}\]
Similarly, for the third term, you raise the common ratio to the second power, and so forth:
\[a_{3}= a_{1} \times r^{2}\]
By methodically applying this technique, each term of the sequence can be determined based on its position in the sequence. This systematic approach is especially useful for longer sequences, where manual calculation of each term would be impractical.

A geometric progression is another term for a geometric sequence. It describes the manner in which the numbers progress from one term to the next, constantly multiplying by the common ratio. The progression can decrease (if \(0 < r < 1\)) or increase (if \(r > 1\)) and can also alternate between positive and negative values if the common ratio is negative, as exhibited in the exercise's sequence. Here, the common ratio of \(-\frac{1}{\(sqrt{2}\)}\) causes the terms to alternate their signs. It's a useful concept not just in pure mathematics but also in various applications like calculating compound interest, population growth, and much more. Understanding geometric progression is essential for solving a broad spectrum of problems that feature repeated multiplicative processes.