The Power of Geometric Progression: Summing up to 30 Terms
Geometric progression, a fundamental concept in mathematics, has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will explore the sum of the first 30 terms of a geometric progression, starting with 1, 2, 4, 8, 16, and so on.
What is a Geometric Progression?
A geometric progression is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed non-zero constant. The formula to calculate the nth term of a geometric progression is:
an = ar^(n-1)
where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
The Sum of a Geometric Progression
The sum of a finite geometric progression can be calculated using the following formula:
S = a(1 - r^n) / (1 - r)
where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Calculating the Sum of the First 30 Terms
Let's calculate the sum of the first 30 terms of the geometric progression starting with 1, 2, 4, 8, 16, and so on. The common ratio (r) in this case is 2.
Term 1: 1 Term 2: 2 (1 × 2) Term 3: 4 (2 × 2) Term 4: 8 (4 × 2) Term 5: 16 (8 × 2) ... Term 30: 536,870,912 (previous term × 2)
Using the formula for the sum of a geometric progression:
S = 1(1 - 2^30) / (1 - 2)
S = 1,073,741,823
The sum of the first 30 terms is 1,073,741,823.
Applications of Geometric Progression
Geometric progressions have numerous applications in various fields, including:
- Physics: Modeling population growth, chemical reactions, and electrical circuits.
- Engineering: Designing digital circuits, signal processing, and control systems.
- Computer Science: Algorithm design, data compression, and cryptography.
In conclusion, the sum of the first 30 terms of the geometric progression starting with 1, 2, 4, 8, 16, and so on is 1,073,741,823. Geometric progressions have far-reaching implications in various fields, and understanding their properties is essential for analyzing and modeling real-world phenomena.
