Infinite Geometric Series Formula

About Infinite 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). [Inifinite Geometric Series Formula]

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 the common ratio of every two successive terms
• n is the number of terms.
• Formula: The sum formula of a finite geometric series a + ar + ar² + ar³ + ... + a r^n-1 is
• Sum of n terms = a (1 - rⁿ) / (1 - r) (or) a (rⁿ - 1) / (r - 1)
• Here,a is first term
• r is 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 first term
• r is common ratio every two successive terms

What do you mean by Infinite Geometric Series Formula?

• To find the sum of a series that extends to infinity, use the sum of infinite geometric series formula. The sum of infinite GP is another name for this. Even though the series has infinite terms, we discover that the sum of a GP converges to a value. If -1<r<1, the infinite series formula is as follows:
  • Sum = a/(1-r)
  • Here, a is first term of the series, r is common ratio between two consecutive terms and −1 < r < 1
  • Note: If r > 1, sum does not exist as sum does not converge.
  • Sum of an infinite arithmetic sequence is ∞, if d > 0, or
  • Sum of an infinite arithmetic sequence is ∞, if d > 0- ∞, if d < 0.

Example: Using an infinite series formula, find the sum of infinite series: 1/4 + 1/16 + 1/64 + 1/256 +?

• Sol: It is given that a = ¼ and r = (1/16) / (1/4) = (1/64) / (1/16) = ¼
• To find out the Sum of given infinite series If r<1 is then sum is given as Sum = a/(1-r)
• Applying the values to the infinite series formula, we get
• Sum = (1⁄4)/(1-1⁄4)
• Sum=(1⁄4)/(3⁄4)
• Sum = 4/(3 × 4)
• Sum = 1/3
• Answer: The sum of 1/4+1/16+1/64+1/256 +? is 1/3

Maths Formulas prepared by HT experts are listed on the main page.

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