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In mathematics, sequences and series are foundational concepts that students must master. A sequence is essentially a list of numbers generated by some explicit rule or function. This rule dictates the output for each position in the sequence, which we then call terms. A term in a sequence can be derived from the previous term or terms in the sequence, or through a general term that describes every term of the sequence with respect to its position.

For example, the sequence from the exercise, where the general term is given by \(a_n = (-3)^n\), is a clear representation of how sequences work. Each term is calculated based on its position \(n\) using a specific mathematical operation. On the other hand, series are about summing the terms of a sequence. Instead of just listing numbers, you'd be adding them up. For the sequence given, if we were to create a series, it would involve the sum of the terms \(a_1, a_2, a_3, ...\), and so on.

A specific type of sequence is called a geometric sequence. Each term in a geometric sequence is found by multiplying the previous term by a constant, which is called the common ratio. The sequence described in the exercise, \(a_n = (-3)^n\), is an example of a geometric sequence where the common ratio is \( -3 \).

To identify if a sequence is geometric, you can divide a term by the preceding term. If this ratio remains constant throughout, then it is indeed a geometric sequence. For instance, dividing the second term \(a_2 = 9\) by the first term \(a_1 = -3\), we get \( -3 \), which is the common ratio here. Repeating this process with \(a_3 / a_2\) should also yield the same common ratio, confirming the geometric nature of the sequence.

Exponents and powers are used to express repeated multiplication of the same number by itself. In the given exercise \(a_n = (-3)^n\), the exponent is \(n\), which indicates the number of times \( -3 \), the base, is multiplied by itself.

The first term \(a_1 = (-3)^1\) simply means \( -3 \), but for the second term, \(a_2 = (-3)^2\), \( -3 \), the base, is multiplied by itself once which results in 9. An important property to remember with exponents is that when the base is negative and the exponent is an even number, the result is positive; when the exponent is odd, the result is negative. This is visible in the alternating positive and negative values of the terms in the problem's sequence.