Calculating the Finite Sum of a Geometric Sequence: A Step-by-Step Guide with Example Calculation

The Geometric Sequence Finite Sum

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

The finite sum of a geometric sequence refers to finding the sum of a specific number of terms in the sequence. In order to calculate the finite sum, we need to know the first term (a) of the sequence, the common ratio (r), and the number of terms (n) we wish to add.

The formula for calculating 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,
a represents the first term of the sequence,
r represents the common ratio between consecutive terms,
and n represents the number of terms.

To understand this formula better, let’s go through an example:

Example:
Find the sum of the first 5 terms of the geometric sequence where the first term is 2 and the common ratio is 3.

Solution:
In this example, a = 2 (first term), r = 3 (common ratio), and n = 5 (number of terms).

Using the formula mentioned earlier, we can calculate the sum of the first 5 terms:

S_n = a * (1 – r^n) / (1 – r)
= 2 * (1 – 3^5) / (1 – 3)
= 2 * (1 – 243) / (-2)
= 2 * (-242) / (-2)
= -484 / (-2)
= 242

Therefore, the sum of the first 5 terms of the geometric sequence with a first term of 2 and a common ratio of 3 is equal to 242.

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