The Geometric Sequence Finite Sum
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero constant called the common ratio (denoted by “r”)
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero constant called the common ratio (denoted by “r”). The general form of a geometric sequence is represented as:
a, ar, ar^2, ar^3, …
Where “a” is the first term and “r” is the common ratio.
The finite sum of a geometric sequence refers to the sum of a specified number of terms in the sequence. To find the finite sum of a geometric sequence, there is a formula based on the first term, common ratio, and the number of terms in the sum.
The formula for the finite sum of a geometric sequence is:
S_n = a * (1 – r^n) / (1 – r)
Where:
– S_n represents the sum of the first n terms of the geometric sequence.
– a is the first term of the geometric sequence.
– r is the common ratio of the geometric sequence.
– n is the number of terms being summed.
Let’s take an example to illustrate the concept. Consider a geometric sequence with a first term of 2 and a common ratio of 3. We want to find the sum of the first 5 terms in the sequence.
Using the formula:
S_5 = 2 * (1 – 3^5) / (1 – 3)
Simplifying the equation:
S_5 = 2 * (1 – 243) / (1 – 3)
S_5 = 2 * (-242) / (-2)
S_5 = 484
Therefore, the sum of the first 5 terms in this geometric sequence is 484.
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