Sum Of First ‘n’ Terms Of An Geometric Progression(G.P) Calculator

$\color{red} \bold{Related \space Calculators:}$
Geometric Mean
nth term of a Arithmetic Progression
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Sum of first n terms of an A.P.
Sum of infinite terms of a G.P.

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

• 1. Introduction to the Sum of the first n terms of a Geometric progression
• 2. What is the Formulae used ?
• 3. How do I calculate the Sum of the first n terms of a Geometric progression ?
• 4. Why choose our Sum of the first n 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 first n terms of a Geometric progression

In mathematics, geometric progressions (often abbreviated as GP) hold a unique place. These sequences appear in various aspects of life, from financial growth to population dynamics. Knowing how to calculate the sum of the first n terms of a geometric progression is a valuable skill with broad applications. In this blog, we will embark on a journey to understand the sum of a geometric progression, explore the formula used, learn how to calculate it, work through examples, address common questions, examine real-life applications, and conclude with an appreciation for the versatility of this mathematical aspect.
$\bold{Definition}$
A geometric progression is a sequence of numbers in which each term is obtained by multiplying the preceding term by a fixed number called the common ratio, denoted as 'r.' The sum of the first n terms of a geometric progression is the total obtained by adding the first n terms of the sequence.

2. What is the Formulae used?

The formula for finding the Sum of the first n terms of a Geometric progression is given by:
$\bold{S_n}$ = $\bold{a.\frac{(r^n - 1)}{(r - 1)}}$
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 & $\bold{S_n}$ represents the Sum of the first n terms of a geometric progression.

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

Identify the value of a, r, and n.
Use the above formula to calculate the Sum of the first n terms of a Geometric progression.

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

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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.

$\bold{Versatility}$
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 how to find the Sum of the first n terms of a Geometric progression.

6. How to use this calculator

This calculator will help you find the sum of the first n terms of a geometric progression.
In the given input boxes, you must put the value of a, r, and n.
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 first 5 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, and n = 5
Since the value of r > 1 it means it is an increasing sequence.
By using the above formula, Sum = $\bold{2.\frac{(3^5 - 1)}{(3 - 1)}}$ = 242
So, the sum of the series is 242.

$\bold{Question:2}$
Determine the sum of the first 8 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 is a decreasing sequence.
By using the above formula, Sum = $\bold{10.\frac{(0.5^8 - 1)}{(0.5 - 1)}}$ = 19.921875
So, the sum of the series is 19.921875.

8. Frequently Asked Questions (FAQs)

What are some real-life applications of geometric progressions?

GP is used in finance for compound interest calculations, in biology for population growth modeling, and physics for exponential decay processes.

Can the formula be used for a decreasing geometric progression?

Yes, it works for both increasing and decreasing GPs, as long as the common ratio (r) is not zero.

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

In that case, the GP becomes a sequence of identical terms, and the sum of the terms is simply n times the first term (a).

Are there other methods to calculate the sum of a GP?

Yes, you can use recursive formulas or derive the sum from the nth-term formula, but the formula mentioned here is the most direct method.

How does a geometric progression differ from an arithmetic progression?

The difference between consecutive terms is constant in an arithmetic progression, whereas the ratio between successive terms is constant in a geometric progression.

9. What are the real-life applications?

The concept of geometric progressions finds practical applications in various fields, including finance (compound interest), biology (population growth), and physics (decay processes).

10. Conclusion

The sum of the first n terms of a geometric progression is a versatile mathematical concept that has far-reaching applications in the real world. With the formula and the ability to calculate it, you possess a powerful tool for analyzing exponential growth or decay phenomena and making informed decisions in various fields. Through concepts like this, mathematics continues to empower us with the tools needed to understand and navigate the complexities of our world.

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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