How to Calculate the Sum of a Geometric Series

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In mathematics, a sequence is any string of numbers arranged in increasing or decreasing order. A sequence becomes a geometric sequence when you are able to obtain each number by multiplying the previous number by a common factor. For example, the series 1, 2, 4, 8, 16 . . . is a geometric sequence with the common factor 2. If you multiply any number in the series by 2, you'll get the next number. By contrast, the sequence 2, 3, 5, 8, 14, 22 . . . is not geometric because there's no common factor between numbers. A geometric sequence can have a fractional common factor, in which case each successive number is smaller than the one preceding it. 1, 1/2, 1/4, 1/8 . . . is an example. Its common factor is 1/2.

The fact that a geometric sequence has a common factor allows you to do two things. The first is to calculate any random element in the sequence (which mathematicians like to call the "​n​th" element), and the second is to find the sum of the geometric sequence up to the ​n​th element. When you sum the sequence by putting a plus sign between each pair of terms, you turn the sequence into a geometric series.

Finding the nth Element in a Geometric Series

In general, you can represent any geometric series in the following way:

a + ar + ar^2 + ar^3 + ar^4 + . . .

where "​a​" is the first term in the series and "​r​" is the common factor. To check this, consider the series in which ​a​ = 1 and ​r​ = 2. You get 1 + 2 + 4 + 8 + 16 . . . it works!

Having established this, it's now possible to derive a formula for the nth term in the sequence (​xn).

x_n = ar^{(n-1)}

The exponent is ​n​ − 1 rather than ​n​ to allow for the first term in the sequence to be written as ​ar0, which equals "​a​."

Check this by calculating the 4th term in the example series.

x_4 = (1) × 2^3 = 8

Calculating the Sum of a Geometric Sequence

If you want to sum a divergent sequence, which is one with a common ration greater than 1 or less than -1, you can can only do so up to a finite number of terms. It is possible to calculate the sum of an infinite convergent sequence, however, which is one with a common ratio between 1 and − 1.

To develop the geometric sum formula, start by considering what you're doing. You're looking for the total of the following series of additions:

a + ar + ar^2 + ar^3 + . . . + ar^{(n-1)}

Each term in the series is ​ark​, and ​k​ goes from 0 to ​n​ − 1. The formula for the sum of the series makes use of the capital sigma sign – ∑ – which means to add all terms from (​k​ = 0) to (​k​ = ​n​ − 1).

\sum_k^{n-1} ar^k = a\bigg(\frac{1 - r^n}{1 - r}\bigg)

To check this, consider the sum of the first 4 terms of the geometric series starting at 1 and having a common factor of 2. In the above formula, ​a​ = 1, ​r​ = 2 and ​n​ = 4. Plugging in these values, you get:

1 \bigg(\frac{1 - 2^4}{1 - 2}\bigg) = 15

This is easy to verify by adding the numbers in the series yourself. In fact, when you need the sum of a geometric series, it's usually easier add the numbers yourself when there are only a few terms. If the series has a large number of terms, though, it's far easier to use the geometric sum formula.

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