Geometric Series

The sum of the terms in a geometric sequence is called a geometric series. The process for computing the sum of the terms in a geometric series is different from the method for solving the arithmetic series.

Find the sum: 1 + 2 + 4 + 8 + 16 + … + 2048

Let S represent the sum of the series.

S = 1 + 2 + 4 + 8 + 16 + … + 2048

Multiply both sides of the equation by the common ratio. It is convenient to offset the product (as shown here).

S = 1 + 2 + 4 + 8 + 16 + … + 2048

2S = 2 + 4 + 8 + 16 + … + 2048 + 4096

Subtract the two equations. (Observe how the middle terms drop out of the problem.)

-S = 1 − 4096

S = 4095

The function defining the terms in this series is

. The formula for finding the sum is:

, where a1 represents the first term in the series.

Using summation notation, the sum can be computed as

.

Find the sum of the first 40 terms of the geometric series whose terms are determined by the formula

.

Using the summation notation yields

. By this formula,
.

Find the sum of the first 25 terms in the geometric series 4 + 12 + 36 + 108 + ….

The common ratio is 3. f(1) = 4 = a(3) implies that a = 4/3. The function that generates the terms of the series is

. Using summation notation, you find that the sum is
, and by this formula,
.

You notice that this is a very large number, and although the previous example had more terms, the sum was very close to 3600. The base of

is a number smaller than 1. The larger the domain value applied to this function, the smaller the term gets, and the less it adds to the accumulated sum. In fact, if the absolute value of the base of the exponential function is less than 1, the sum of an infinite number of terms will reach a finite value.

The sum of an infinite geometric series is

when |r| < 1.

Find the sum: 2000 + 1000 + 500 + 250 + …

The first term of the series is 2000, and the common ratio is 1/2. Therefore, the sum is:

.

Find the value of

.

The first term is f(1) = (-18)(-2/3) = 12, and the common ratio is −2/3. Therefore, the sum of the infinite number of terms for this series is

.

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