A geometric progression (GP) is a powerful mathematical sequence where each term is found by multiplying the previous term by a constant value called the common ratio. Understanding the sum of infinite GP and related concepts is crucial for advanced mathematics.
When working with a geometric sequence, we often need to calculate the sum of n terms of GP for various applications. A geometric progression takes the form:
a, ar, ar², ar³, ar⁴...
Where:
The sum of n terms of GP formula provides the total when adding a finite number of terms:
S₍ₙ₎ = a(1-rⁿ)/(1-r) (when r ≠ 1)
If r = 1, then the sum of n terms of GP simplifies to S₍ₙ₎ = na
The infinite GP formula represents one of mathematics' most elegant concepts. The sum of infinite terms of GP follows a specific pattern with an important condition: the series only converges when |r| < 1. When this condition is met:
Infinite GP formula: S∞ = a/(1-r)
This sum of infinite GP series expression is remarkably simple yet powerful in solving countless problems.
For an infinite GP sum to exist as a finite value, |r| < 1 is required. Under this condition, each term becomes progressively smaller, ensuring the sum of infinite geometric series converges. When |r| ≥ 1, the sum of infinite terms of GP becomes infinite or undefined.
Looking at 1/2 + 1/4 + 1/8 + ...
For 3 + 0.3 + 0.03 + ...
The infinite GP sum concept appears in various fields:
When r is negative, we have an alternating series. For example: 1 - 1/2 + 1/4 - 1/8 + ...
Sometimes we encounter series that don't start with the first term: ar + ar² + ar³ + ... This equals r times the sum of infinite geometric series: S∞ = r·a/(1-r) = ar/(1-r)
The elegant simplicity of the infinite GP formula makes it one of the most useful tools in mathematics, allowing us to find exact values for sum to infinity of GP that appear to go on forever.
To master the sum of infinite geometric series, try these practice problems:
Understanding the infinite GP formula and its applications will strengthen your mathematical toolkit and problem-solving abilities across various disciplines
A geometric progression is a sequence where each term is found by multiplying the previous term by a fixed non-zero number called the common ratio. For example, 2, 6, 18, 54, ... is a GP with first term 2 and common ratio 3.
The sum of n terms of GP refers to adding together a finite number of terms in a geometric progression. The formula for the sum of n terms of GP is: S₍ₙ₎ = a(1-rⁿ)/(1-r) when r ≠ 1 S₍ₙ₎ = na when r = 1 Where a is the first term and r is the common ratio.
The infinite GP formula calculates the sum of infinite terms of GP when the series converges. The formula is: S∞ = a/(1-r) when |r| < 1 This formula represents the sum to infinity of GP in its simplest form.
The sum of infinite GP exists only when the absolute value of the common ratio is less than 1 (|r| < 1). This ensures that the terms become progressively smaller, allowing the sum of infinite geometric series to converge to a finite value.
To calculate the sum of infinite GP:
The sum of n terms of GP uses the formula S₍ₙ₎ = a(1-rⁿ)/(1-r) for a finite number of terms, while the sum to infinity of GP uses S∞ = a/(1-r) when |r| < 1. The infinite formula is simpler because as n approaches infinity, rⁿ approaches 0 when |r| < 1.
Yes, the sum of infinite GP series can be negative. This occurs when either:
When r = -1, the infinite GP sum does not converge because the terms alternate between two values without getting smaller. The formula for sum of infinite terms of GP cannot be applied in this case.
Example: Find the sum of infinite GP: 6 + 2 + 2/3 + 2/9 + ...
Example: Calculate the sum to infinity of GP: 5 + 0.5 + 0.05 + ...
Recurring decimals are perfect examples of infinite geometric series. For example: 0.999... = 9/10 + 9/100 + 9/1000 + ... This is an infinite GP with a = 9/10 and r = 1/10 Using the sum of infinite GP formula: 0.999... = (9/10)/(1-1/10) = (9/10)/(9/10) = 1
The sum of infinite GP series has many practical applications:
A convergent infinite GP is one where the sum of infinite terms of GP approaches a fixed, finite value. This occurs when |r| < 1, causing each successive term to be smaller than the previous one, allowing the sum to infinity of GP to reach a limit.
A divergent infinite GP is one where the sum of infinite terms of GP does not approach a finite value. This occurs when |r| ≥ 1. In these cases, the infinite GP formula S∞ = a/(1-r) cannot be applied.
A geometric series is the sum of terms in a geometric progression. The sum of infinite geometric series refers specifically to adding all terms in a GP that continues indefinitely. When the series converges (|r| < 1), we can apply the infinite GP formula to find its exact value.
Yes, the sum of infinite GP formula can be extended to complex numbers. The condition for convergence remains |r| < 1, where |r| represents the modulus (absolute value) of the complex number r. The formula S∞ = a/(1-r) still applies.
If your infinite GP sum doesn't converge, check if |r| ≥ 1. The sum of infinite terms of GP only converges when |r| < 1. Common mistakes include:
Common mistakes when calculating the sum of infinite GP series include:
An alternating infinite GP has terms that switch between positive and negative. This occurs when r < 0. To find the sum of infinite geometric series that alternates:
For example, for 8 - 4 + 2 - 1 + ..., a = 8, r = -1/2, so S∞ = 8/(1-(-1/2)) = 8/1.5 = 16/3.
If your infinite GP starts with a term other than the first term (like ar², ar³, etc.), you can: