Calculating the Finite Sum of a Geometric Sequence | Formula and Example

The Geometric Sequence Finite Sum

The finite sum of a geometric sequence refers to the sum of a specific number of terms in the sequence

The finite sum of a geometric sequence refers to the sum of a specific number of terms in the sequence.

Before discussing the formula for calculating the finite sum, it is important to understand what a geometric sequence is. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (denoted by ‘r’).

The general form of a geometric sequence is:
a, ar, ar^2, ar^3, ar^4, …

In this sequence, ‘a’ represents the first term, and ‘r’ represents the common ratio.

To find the finite sum of a geometric sequence, we can use the following formula:

Sn = a * (1 – r^n) / (1 – r)

In this formula, ‘Sn’ represents the sum of the first ‘n’ terms, ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the number of terms that we want to sum.

Let’s consider an example to illustrate the concept:
Find the sum of the first 5 terms in the geometric sequence with a first term of 2 and a common ratio of 3.

Using the formula, we have:
S5 = 2 * (1 – 3^5) / (1 – 3)

Simplifying the formula:

S5 = 2 * (-242) / (-2)
S5 = 484

Therefore, the sum of the first 5 terms in this geometric sequence is 484.

It is important to note that the formula we used assumes that the common ratio ‘r’ is not equal to 1. If ‘r’ equals 1, it means that the sequence is not actually a geometric sequence, and the formula does not apply. In that case, to calculate the sum of the first ‘n’ terms, you simply multiply the first term ‘a’ by ‘n’.

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