Understanding the Infinite Sum of a Geometric Sequence: Formula, Calculation, and Examples.

The Geometric Sequence Infinite Sum

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

To understand the infinite sum of a geometric sequence, we need to first 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 preceding term by a fixed, non-zero number. This fixed number is called the common ratio, denoted by ‘r’.

For example, a geometric sequence could be: 1, 2, 4, 8, 16, …

In this sequence, the common ratio ‘r’ is 2, because each next term is found by multiplying the previous term by 2.

The formula to find the nth term of a geometric sequence is:

an = a1 * r^(n-1)

Where:
an = the nth term
a1 = the first term
r = the common ratio
n = the position of the term

Now, let’s focus on finding the sum of an infinite geometric sequence. An infinite geometric sequence is a sequence that goes on indefinitely.

The formula to find the infinite sum (sum to infinity) of a convergent geometric sequence is:

S = a1 / (1 – r)

Where:
S = the sum to infinity
a1 = the first term
r = the common ratio

This formula holds true only when the common ratio ‘r’ is between -1 and 1.

If the absolute value of the common ratio (|r|) is greater than or equal to 1, then the sequence diverges and the infinite sum does not exist.

Let’s illustrate this with an example:

Consider the geometric sequence: 2, -4, 8, -16, …

In this sequence, the first term (a1) is 2 and the common ratio (r) is -2.

To find the infinite sum (S), we can use the formula:

S = a1 / (1 – r)

Substituting the values, we get:

S = 2 / (1 – (-2))
S = 2 / (1 + 2)
S = 2 / 3

So, the infinite sum of the given geometric sequence is 2/3.

It is important to note that this formula only works for convergent geometric sequences with a common ratio between -1 and 1. For divergent sequences or common ratios outside this range, the infinite sum does not exist.

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