Solving the Geometric Progression Series: 1/8, 1/8^2, 1/8^3, 1/8^4, 1/8^5
When dealing with sequences in mathematics, understanding geometric progressions is essential. This article provides a detailed guide on how to solve the series given by 1/8, 1/82, 1/83, 1/84, 1/85. We will explore the properties of geometric progressions and apply the formula to find the sum of this series.
Understanding Geometric Progressions (GPs)
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to sum the first n terms of a geometric progression is given by:
Sn a(1 - rn) / (1 - r)
where u03B1 (a) is the first term, r is the common ratio, and n is the number of terms.
Solving the Given Series 1/8, 1/82, 1/83, 1/84, 1/85
Let's start by defining the series in terms of a geometric progression. The first term (a) is 1/8, and the common ratio (r) is also 1/8 since
r 1/8^2 / 1/8 1/8
Given that we have 5 terms (n 5), we can now apply the formula to find the sum:
S5 (1/8) (1 - (1/8)5) / (1 - 1/8)
Step-by-Step Calculation
First, calculate the numerator:1 - (1/8)5 1 - 1/32768 32767/32768
Calculate the denominator:1 - 1/8 7/8
Multiply the results:(1/8) * (32767/32768) / (7/8) (32767/32768) / (7/8) (32767/32768) * (8/7)
Final result:32767 * 8 / (32768 * 7) 262136 / 229376 4681/32768
Conclusion
The sum of the series 1/8, 1/82, 1/83, 1/84, 1/85 is thus 4681/32768. This solution highlights the application of geometric progression concepts in solving real mathematical series problems. Understanding these principles is crucial for further exploration into more complex mathematical fields.
Additional Resources
Learn more about geometric progressions on Wikipedia