What is the formula for the sum of the first n terms in a geometric sequence?

The formula for the sum of the first n terms in a geometric sequence is indeed represented as (a*(1-r^n))/(1-r), where 'a' is the first term of the sequence, 'r' is the common ratio, and 'n' is the number of terms.

When deriving this formula, we recognize that a geometric series is generated by multiplying the preceding term by a fixed number (the common ratio, r). In this context, the terms of the geometric sequence can be expressed as a, ar, ar^2, ar^3, ..., ar^(n-1).

To compute the sum of these terms, we can use the property of geometric series. If we denote the sum S of the first n terms, it can be expressed as S = a + ar + ar^2 + ... + ar^(n-1). By multiplying the entire sum by the common ratio 'r', we create a new equation: S * r = ar + ar^2 + ar^3 + ... + ar^n.

When we subtract this new equation from the original sum S, most terms cancel out, leading to a simplified form. This simplification allows us to isolate S and ultimately arrive at the