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A geometric series is a sum of terms where each term is a fixed multiple of the previous one. In a geometric progression, you will often hear the term 'common ratio,' which determines how the series grows or shrinks. When you have a series like 1, 2, 4, ..., each term is obtained by multiplying the previous term by a fixed number.

A geometric series helps in calculating a sum of such sequences where this regular multiplication occurs. This makes geometric series especially handy in many practical applications like calculating interest in finance, computer algorithms, and solving physics problems.

The common ratio in a geometric sequence is the number you multiply by to get from one term to the next. It's what makes the series geometric. In our series, 1, 2, 4, ..., the common ratio is 2. So, if the sequence is 1, 2, 4, ..., each term during this progression results from multiplying the previous term by the common ratio of 2.

To find the common ratio, you can take any term and divide it by the previous term. In this sequence, dividing 2 by 1 or 4 by 2 always gives 2, hence confirming that 2 is indeed the common ratio.

To find the sum of a geometric series, there is a neat formula that simplifies the process. The sum formula for a finite geometric series is: \[ S_n = a \frac{(1 - r^n)}{1 - r} \]Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. You simply plug in your values and calculate the sum.For instance, in our sequence of 1 + 2 + 4 + ..., applying the formula where \(a = 1\), \(r = 2\), and \(n = 8\), it becomes \( S_8 = 1 \frac{(1 - 2^8)}{1 - 2} \). This expression gives you the total of all terms in the sequence, which turns out to be 255 after simplifying.

Unlike geometric series, an arithmetic series is another type of sequence where the difference between consecutive terms remains constant, known as the 'common difference.'.For example, in an arithmetic series like 3, 6, 9, 12, ..., each number is increased by a common difference, such as 3. This is different from a geometric series where you multiply by a common ratio.Though not applied in the original exercise, knowing the distinction between arithmetic and geometric series clarifies various problems. In arithmetic series, you can also find a sum, but the formula differs: \[ S_n = \frac{n}{2} \times (a + l) \]. Here, \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.