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In algebra, sequences are a list of numbers following a certain rule. Algebraic sequences can be linear or nonlinear, and one special type of nonlinear sequence is known as a geometric sequence. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number, often referred to as the 'common ratio'.

For example, in the sequence 2, 4, 8, 16, ..., each term doubles the one before it, making it a geometric sequence with a common ratio of 2. Distinguishing the type of sequence is essential because it guides us to use the correct formulae to find subsequent terms, sums of certain number of terms or even the nth term. To understand and work with algebraic sequences, we need to grasp the concept of common ratio in the case of geometric sequences and common difference in arithmetic sequences.

Finding the nth term of a sequence involves using a formula that provides the value of any term based on its position within the sequence. For a geometric sequence, the nth term can be found using the formula: \(a_n = a_1 \times r^{n-1}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term's position.

This formula allows us to quickly calculate the value of a term far down the sequence without the need to write out all the previous terms. For example, to find the 7th term of the sequence 11, 33, 99, and so on, we identify the first term (11) and the common ratio (3), then apply them to the formula \(a_7 = 11 \times 3^{7-1}\) to obtain 8019. Understanding how to apply this formula is crucial for solving problems related to geometric sequences.

An exponential expression is a mathematical notation that indicates the operation of raising a number to a power. In the context of geometric sequences, this operation is used to describe how each term in the sequence is derived from the previous one by multiplying it by a constant base raised to a certain exponent.

The general form of an exponential expression is \(b^n\), where \(b\) is the base and \(n\) is the exponent, signifying that \(b\) is multiplied by itself \(n\) times. For instance, in the formula for the nth term of a geometric sequence \(a_n = a_1 \times r^{n-1}\), the expression \(r^{n-1}\) is an exponential expression where \(r\) is raised to the power of \(n-1\). When calculating the 7th term of our example sequence, we simplify an exponential expression: \(3^6\), which equals 729. Knowledge of how to manipulate exponential expressions is vital when working with sequences, particularly geometric ones.