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A geometric sequence is a mathematical pattern characterized by a constant ratio between consecutive terms. This ratio is called the common ratio and can be any real number. To recognize a geometric sequence, look for a pattern where each term is obtained by multiplying the previous term by a fixed number.

In the provided exercise, the sequence is given by the formula \( a_{n}=\left(-\frac{2}{3}\right)^{n} \), where \( n \) represents the term's position in the sequence. Here, the common ratio is \( -\frac{2}{3} \) since each term is achieved by multiplying the previous term by \( -\frac{2}{3} \). Geometric sequences like this one can alternate in sign if the common ratio is negative, which is the case here.

An exponential expression is a mathematical representation that describes the repeated multiplication of the same factor. It is written in the form \( b^{n} \), where \( b \) is the base and \( n \) is the exponent or power. The exponent dictates how many times the base is used as a factor in the multiplication.

In our sequence example, the term \( \left(-\frac{2}{3}\right)^{n} \) is an exponential expression with a base of \( -\frac{2}{3} \) and an exponent \( n \). When the base is negative, as it is here, the sign of the terms will alternate depending on whether the exponent is odd (negative result) or even (positive result). This alternating pattern is a key feature in understanding how the sequence behaves.

Mathematical induction is a powerful technique used to prove statements or formulas that apply to all natural numbers. It consists of two main steps: the base case and the induction step. First, you prove the statement for the initial value, usually \( n = 1 \), this is called the base case. Then, assuming the statement holds for \( n \) (the inductive hypothesis), you prove that it holds for \( n+1 \) as well, known as the induction step.

While this exercise did not require the use of mathematical induction to solve, understanding this method can be crucial when dealing with sequences, especially when you want to prove general formulas or properties for all terms in a sequence. For instance, were you asked to prove that the general term of our geometric sequence is always less than a certain value, you could use induction to affirm this for all natural numbers \( n \) in the sequence.