Geometric Series Formula

About Geometric Series Formula

Let us first define what a geometric series is before learning the formula. The ratios of every two consecutive terms are the same in this series (sum of terms). The formula for the geometric series is:

1. The nth term of a geometric series can be found using this formula.
2. Calculating the sum of finite geometric series
3. Calculating the total of an infinite geometric series

What do you mean by Geometric Series?

A geometric series is the sum of a geometric sequence's finite or infinite terms. The corresponding geometric series for the geometric sequences a, ar, ar²,..., ar^n-1,... is a + ar + ar² +..., ar^n-1 +.... "Series" means "sum," as we all know. The geometric series is defined as the sum of phrases with a common ratio between every two adjacent terms. Geometric series come in two varieties: finite and infinite. Here are some geometric series examples.

1/2 + 1/4 +.... + 1/8192 is a finite geometric series in which the first term, a = 1/2, and the common ratio, r = 1/2
-4 + 2 - 1 + 1/2 - 1/4 +... is an infinite geometric series in which the first term, a = -4, and the common ratio, r = -1/2.

Geometric Series Formula

The formula for the sum of a finite geometric sequence, the sum of an infinite geometric sequence, and the nth term of a geometric sequence is known as the geometric series formula. The sequence is a, ar, ar², ar³,......, with a being the first term and r being the "common ratio."

• a_n = a · r^{n - 1}
• S_n = a (rⁿ- 1) / (r - 1).
• S_n = a (1 - rⁿ) / (1 - r).
• S_∞ = a / (1 - r).

Geometric Series Formulas
The formulas for finding the nth term, the sum of n terms, and the sum of infinite terms are among the formulas for a geometric series. Consider a geometric series with a first term and r as the common ratio.

• a + ar + ar² + ar³ + ...

Formula : The nth term of a geometric sequence is,nth term = a r^n-1 Here, a is the first term

• r is common ratio of every two successive terms
• n is number of terms.

Formula : The sum formula of a finite geometric series a + ar + ar²+ ar³+ ... + a r^n-1is
Sum of n terms = a (1 - rⁿ) / (1 - r) (or) a (rⁿ- 1) / (r - 1)

Here,

• a is the first term
• r is the common ratio every two consecutive terms
• n is the number of terms.

Formula : The sum formula of an infinite geometric series a + ar + ar²+ ar³+ ..........is
Sum of infinite geometric series = a / (1 - r)

Here,

• a is the first term
• r is the common ratio every two successive terms

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