A geometric series is a succession of numbers in which each term is fetched by multiplying the previous term by a fixed number called the common ratio. The prevalent form of this kind of sequence of numbers can be indicated as:
a, ar, ar^2, ar^3,
where a is the first term in the string and r is the common ratio.
The sum of a finite geometric series is expressed by the formula:
S = a(1-r^n)/(1-r)
where S represents the sum of the series, a represents the first term in the series, r is the common ratio, and n represents the number of terms in the series.
As an example, take the geometric series 2, 6, 18, 54, … into consideration. Here 2 is the first term in the series, and 3 is the common ratio (each term is obtained by multiplying the previous term by 3). The sum of the first 4 terms in the series will be:
S = 2(1-3^4)/(1-3) = 2(-32)/(-2) = -16Therefore, the sum of the first 4 terms in the geometric series 2, 6, 18, 54, as mentioned above is -16.
The geometric series is significant in mathematics and has several real-world uses, that includes finance, and engineering, where quantities grow or decay at a fixed rate.
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