Calculating the Infinite Sum of a Geometric Sequence: Formula and Examples

The Geometric Sequence Infinite Sum

To find the infinite sum of a geometric sequence, we need to understand what a geometric sequence is

To find the infinite sum of a geometric sequence, we need 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 constant ratio.

The general form of a geometric sequence is:

a, ar, ar^2, ar^3, …

Where ‘a’ is the first term and ‘r’ is the common ratio.

To calculate the infinite sum of a geometric sequence, we use the formula:

S = a / (1 – r)

Where ‘S’ represents the sum, ‘a’ is the first term, and ‘r’ is the common ratio. However, for the formula to work, the absolute value of ‘r’ must be less than 1.

Let’s use an example to illustrate this formula:

Example: Find the infinite sum of the geometric sequence 2, 4, 8, 16, 32, …

In this example, the first term ‘a’ is 2, and the common ratio ‘r’ is 2, since each term is obtained by multiplying the previous term by 2.

Using the formula, we have:

S = 2 / (1 – 2)

Since the absolute value of ‘r’ is equal to 2 and is greater than 1, this geometric sequence does not have an infinite sum. In other words, the sum of the terms in this sequence continues to increase indefinitely without converging to a specific value.

However, if we had a geometric sequence with a common ratio ‘r’ that satisfies the condition |r| < 1, then we could use the formula to find the infinite sum. For example, in the sequence 1, 1/2, 1/4, 1/8, ... (where 'a' is 1 and 'r' is 1/2), the infinite sum can be calculated as: S = 1 / (1 - 1/2) Simplifying this expression, we get: S = 1 / (1/2) S = 2 Therefore, the infinite sum of the sequence 1, 1/2, 1/4, 1/8, ... is 2. In conclusion, the infinite sum of a geometric sequence can be found using the formula S = a / (1 - r), as long as the absolute value of 'r' is less than 1.

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