Sum Of Infinite Terms Of A Geometric Progression Calculator

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$\bold{Table \space of \space Content}$

• 1. Introduction to the Sum of the infinite terms of a Geometric progression
• 2. What is the Formulae used ?
• 3. How do I calculate the Sum of the Infinite terms of a Geometric progression ?
• 4. Why choose our Sum of the Infinite terms of a Geometric progression Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Sum of the Infinite terms of a Geometric progression

In the vast landscape of mathematics, few concepts are as fascinating and powerful as the sum of an infinite geometric progression. These sequences grow or shrink exponentially and hold secrets to understanding phenomena from compound interest to population dynamics. Calculating the sum of an endless series can seem daunting, but it's an essential skill with broad applications. In this blog, we'll journey to comprehend the sum of an infinite geometric progression. We will explore the formula and the conditions required for convergence, learn how to calculate it, tackle examples, address common questions, delve into real-life applications, and conclude with a deep appreciation for the elegance of this mathematical concept.
$\bold{Definition}$
An infinite geometric progression is a sequence of numbers. Each term is obtained by multiplying the preceding term by a fixed number called the common ratio, denoted as 'r.' The sum of an infinite geometric progression is the total obtained by adding all the terms of the sequence to infinity.

2. What is the Formulae used?

The formula for finding the Sum of the Infinite terms of a Geometric progression is given by:
$\bold{S_{inf}}$ = $\bold{\frac{a}{1 - r}}$ where $\bold{-1 < r < 1}$.
Where $\bold{a}$ is the first term of the sequence & $\bold{r}$ is the common ratio.
$\bold{n}$ is the number of terms of which sum is required & $\bold{S_{inf}}$ represents the Sum of the Infinite terms of a geometric progression.

$\bold{Conditions \space Required}$
To calculate the sum of an infinite geometric progression using the formula, the common ratio (r) must satisfy the condition $\bold{| r | < 1}$.
The above condition ensures that the series converges to a finite value.
If $\bold{| r |}$ $\ge$ 1, the series diverges, and no finite sum exists.

3. How do I calculate the Sum of the Infinite terms of a Geometric progression?

Identify the value of a and r.
Use the above-given formula to calculate the Sum of the Infinite terms of a Geometric progression.

4. Why choose our Addition/Subtraction of two Vectors Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

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Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

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Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on the concept of how to find the Sum of the Infinite terms of a Geometric progression.

6. How to use this calculator

This calculator will help you find the sum of the infinite terms of a geometric progression.
In the given input boxes, you must put values a and r.
After clicking on the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question:1}$
Find the sum of the infinite terms of a geometric progression with the first term (a) as 2 and the common ratio (r) as 3. $\bold{Solution}$
Given value of a = 2, r = 3
Since the value of r > 1 it means it is an increasing sequence.
By using the above concept, Sum = diverges
So the sum of the series is Diverges.

$\bold{Question:2}$
Determine the sum of the infinite terms of a geometric progression where the first term (a) is 10 and the common ratio (r) is 0.5.
$\bold{Solution}$
Given value of a = 10, r = 0.5, and n = 8
Since the value of r < 1 it means it is a decreasing sequence.
By using the above formula, Sum = $\bold{\frac{10}{1 - 0.5}}$ = 20
So the sum of the series is 20.

8. Frequently Asked Questions (FAQs)

Why is it important to check the condition ∣r∣<1 for convergence?

Ensuring ∣r∣<1 is crucial because it determines whether the series converges to a finite value or diverges to infinity.

What happens if the common ratio (r) exceeds or equals 1?

If ∣r∣≥1, the series diverges, and no finite sum exists. The series keeps growing or oscillating indefinitely.

Can the formula be used to decrease geometric progressions?

Yes, it works for increasing and decreasing GPs as long as ∣r∣<1.

Are there other methods to calculate the sum of an infinite GP?

Yes, you can derive the sum using the concept of limits or by calculating the sum of a finite portion and taking the limit as the number of terms approaches infinity.

Is the sum of an infinite geometric series exact?

No, Sum is not the exact value. Still, approximately, it is equal to the real some because for a decreasing sequence having |r| < 1 as n approaches infinity, the term's value approaches zero.

9. What are the real-life applications?

In finance, the sum of an infinite geometric progression is used to calculate the present value of a series of future cash flows, such as annuities or investment returns. This calculation is essential in determining the current worth of future payments or investments.

10. Conclusion

The sum of an infinite geometric progression is a captivating mathematical concept key to understanding exponential growth and decay phenomena in various fields. By mastering the formula and the conditions for convergence, you acquire a powerful tool for making informed decisions in finance, physics, and beyond. Through concepts like this, mathematics continues to illuminate the world around us, revealing the beauty and utility of its principles.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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