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Understanding the common ratio is crucial when studying geometric sequences. It is the factor by which each term of the sequence is multiplied to get the next term. In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number, known as the common ratio.

For example, in the sequence provided in the exercise, the terms are 2, 6, 18, 54, and so on. To find the common ratio, you divide the current term by the previous one. From the second term (6) divided by the first term (2), we calculate a common ratio of \( r = \frac{6}{2} = 3 \). This ratio is consistent for every consecutive pair of terms in the sequence. If the common ratio is greater than 1, the terms in the sequence will grow larger; if the ratio is between 0 and 1, the terms will get smaller; and if the ratio is negative, the terms will alternate in sign.

The sum of a geometric series is an essential concept for understanding and working with geometric sequences. It allows you to find the total sum of a finite number of terms in the sequence. A formula to compute the sum (\( S_n \)) of the first \( n \) terms of a geometric sequence is given by:

\( S_n = a_1 \frac{1 - r^n}{1 - r} \)

Here, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms to be added. This formula is particularly helpful when the number of terms is large, making manual addition impractical. In our exercise, the sum of the preceding terms is found using this formula but with \( n-1 \) since we want to include all terms before the \(n^{th}\) term. Adding 1 to this sum represents the total sum of 1 and all terms up to the \( (n-1)^{th} \) term.

The geometric sequence formula provides a straightforward method to find any term within a geometric sequence without listing all previous terms. This powerful tool is expressed as:\
\( a_n = a_1 \times r^{n-1} \)\

Where \( a_n \) is the \( n^{th} \) term of the sequence, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. With the formula, calculating any term in the sequence becomes a matter of plugging in the values for \( a_1 \) and \( r \) and raising the common ratio to the power of \( n-1\). For instance, the sequence from our exercise defines \( a_1 \) as 2 and \( r \) as 3, allowing us to calculate any subsequent term easily.

Mathematical induction is a proof technique used to establish the truth of infinite statements or sequences, like verifying properties of a geometric sequence. It consists of two critical steps: showing the base case is true (usually the first term in sequences) and then proving that if the \( (k)^{th} \) statement is true, the next statement \( (k+1)^{th} \) is also true.

In the context of the given exercise, we start by assuming that a certain property holds for the first few terms. Then, using the principle of mathematical induction, we prove that if the property holds for the \( n^{th} \) term, it must naturally hold for the \( (n+1)^{th} \) term. This method not only shows the property is true for all the terms covered in the sequence but also provides a logical and structured approach to tackling problems involving progression.