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The common ratio is a key concept in understanding a geometric series. It is the factor by which each term in the series is multiplied to get the next term. Simply put, if you have any two consecutive terms in a geometric sequence, you get the common ratio by dividing one by the other.

For instance, consider the series \(1, 2, 4, 8, 16, ...\). To find the common ratio, divide the second term by the first (which is 2 ÷ 1) and you get 2. This means each term is twice the previous term. In our exercise, the common ratio is \(r = \frac{4}{3}\), and it is greater than 1, indicating that our series is increasing and each term is \frac{4}{3} times the term before it. Understanding the common ratio is vital because it dictates the growth of the series and is crucial in determining the sum of the series using the sum formula.

The sum formula for a geometric series is an elegant equation that allows us to find the sum of all terms in the series without needing to add each one individually. This is especially useful for series with a large number of terms or those that extend infinitely.

The general form of the sum formula for a geometric series with a ratio different from 1 is: \[ S_n = \frac{a_1(r^n - 1)}{r - 1} \] where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term of the series, \(r\) is the common ratio, and \(n\) is the number of terms. When \(r > 1\), each term is greater than the previous term, and when \(r < 1\), the terms get progressively smaller.

In our textbook exercise, we applied this formula by plugging in the first term \(a_1 = \frac{4}{3}\), the common ratio \(r = \frac{4}{3}\), and the number of terms \(n = 14\) to find the sum of the series. By using the formula correctly, we can conclude that summing up the series is a process that can be performed with a straightforward calculation.

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'. It's a beautifully simple yet powerful concept in mathematics.

For example, in a geometric sequence such as \(2, 4, 8, 16, ...\), the first term is 2, and the common ratio is 2, since each term is 2 times its predecessor. Geometric sequences can model many natural phenomena like exponential growth or decay, such as in population dynamics or radioactive decay.

The sequence from our exercise \(\left(\frac{4}{3}\right)^1, \left(\frac{4}{3}\right)^2, ..., \left(\frac{4}{3}\right)^{14}\) is geometric because each term can be obtained by multiplying the previous term by \(\frac{4}{3}\). Recognizing a sequence as geometric allows us to predict future terms and calculate properties such as the sum of its terms using the sum formula, as we have demonstrated in the exercise.