91Ó°ÊÓ

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, known as the common ratio \(r\). Each number in the progression is called a term. For example, in the sequence \(2, 4, 8, 16, \dots\), we start with the first term \(2\) and multiply by the common ratio \(2\) to get each subsequent term.

If the common ratio is between \(0\) and \(1\), the terms will get increasingly smaller, tending towards zero. Conversely, if the ratio is greater than \(1\), the terms will grow towards infinity. The common ratio can also be negative, which would make the sequence alternate between positive and negative values.

The sum of a geometric series is the total value when all terms of a geometric progression are added together. If we have a geometric progression with the first term \(a\), common ratio \(r\), and \(n\) terms, the sum \(S_n\) of the first \(n\) terms can be computed using a specific formula. The sum is finite if \(r\) is between \(1\) and \(−1\); otherwise, the sum can go to infinity, and we would refer to it as an infinite geometric series.

The series presented in the exercise, \(\sum_{k=0}^{5}(\frac{2}{3})^{k}\), has the terms getting smaller because the common ratio (\frac{2}{3}) is less than \(1\). This means that the sum will converge to a finite number, which is computed by substituting suitable values in the sum formula for a finite geometric series.

The geometric series formula for the sum of the first \(n\) terms is given by \(S_n = \frac{a(1-r^n)}{1-r}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

To apply this formula effectively, one should first identify the values of \(a\), \(r\), and \(n\) in the given geometric series. In the context of the exercise, the first term \(a\) is \(1\) because any number raised to the power \(0\) is \(1\), the common ratio \(r\) is \(\frac{2}{3}\), and the number of terms \(n\) is \(6\). After plugging these values into the formula, we simply carry out the arithmetic operations to find the series sum.