Calculating the Sum of an Infinite Geometric Sequence: Understanding the Formula and Examples

The Geometric Sequence Infinite Sum

The sum of an infinite geometric sequence can be calculated when the common ratio (r) has an absolute value less than 1

The sum of an infinite geometric sequence can be calculated when the common ratio (r) has an absolute value less than 1. The formula for the sum of an infinite geometric series is:

S = a / (1 – r),

where S is the sum, a is the first term, and r is the common ratio.

To understand why this formula works, let’s break it down step by step.

Consider an infinite geometric sequence with the first term a and the common ratio r. The individual terms of the sequence can be written as:
a, ar, ar^2, ar^3, ar^4, …

To find the sum of this sequence, we add up all the terms:
S = a + ar + ar^2 + ar^3 + ar^4 + …

Now, let’s multiply both sides of this equation by r:
rS = ar + ar^2 + ar^3 + ar^4 + ar^5 + …

Subtracting these two equations gives us:
S – rS = a,

which can be simplified to:
S(1 – r) = a.

Dividing both sides of the equation by (1 – r), we get the formula for the sum of an infinite geometric sequence:
S = a / (1 – r).

However, it is important to note that this formula only works when the absolute value of r is less than 1. If the absolute value of r is greater than or equal to 1, the sequence does not have a finite sum.

Let’s consider an example to illustrate the use of this formula. Suppose we have the following infinite geometric sequence:
3, 1, 1/3, 1/9, 1/27, …

In this case, the first term (a) is 3, and the common ratio (r) is 1/3. Since the absolute value of r is less than 1, the sequence has a finite sum. Using the formula, we can find the sum (S) as follows:

S = 3 / (1 – 1/3)
= 3 / (2/3)
= 9/2.

Therefore, the sum of the given infinite geometric sequence is 9/2 or 4.5.

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