Geometric Progression Problem Fifteenth Term, Sum To Infinity, First Term, And Common Ratio

Geometric progressions are a fundamental concept in mathematics, particularly in sequences and series. They involve sequences where each term is multiplied by a constant factor to obtain the next term. This article delves into a geometric progression problem where the fifteenth term is related to the twelfth term, and the sum to infinity is known. We aim to find the first term and the common ratio, as well as the least number of terms needed for the sum of the geometric progression to exceed a certain value. Understanding geometric progressions is crucial not only for academic purposes but also for various real-world applications, such as financial calculations, physics, and computer science. The problem we'll tackle involves using the properties of geometric sequences to derive unknown parameters, showcasing the practical application of mathematical concepts. By working through this problem, readers will gain a deeper understanding of how geometric progressions behave and how to manipulate their formulas effectively. This knowledge is invaluable for anyone studying mathematics or working in fields that rely on mathematical modeling and analysis. This article serves as a comprehensive guide to understanding and solving geometric progression problems, providing step-by-step solutions and explanations to aid in comprehension. Let's dive into the problem and explore the fascinating world of geometric sequences.

Problem Statement

Consider a geometric progression where the fifteenth term is equal to ${ \frac{1}{8} }$ of the twelfth term. Additionally, the sum to infinity of this progression is 5. Our task is twofold:

(a) Determine the first term and the common ratio of this geometric progression.

(b) Find the least number of terms needed for the sum of the geometric progression to exceed a certain value (which will be determined in the solution).

This problem combines two critical aspects of geometric progressions: the relationship between terms and the sum to infinity. By solving this, we can understand how the properties of geometric sequences can be used to find specific parameters and understand the behavior of infinite series. Part (a) focuses on finding the fundamental characteristics of the geometric progression, while part (b) delves into the cumulative behavior of the series, adding an additional layer of complexity. This problem is an excellent exercise in applying the formulas and properties of geometric progressions to solve a comprehensive problem.

(a) Finding the First Term and Common Ratio

To solve this part, we need to use the formulas for the nth term and the sum to infinity of a geometric progression. Let's denote the first term as a and the common ratio as r. The formula for the nth term (${ T_n }$) of a geometric progression is:

${ T_n = ar^{n-1} }$

And the sum to infinity (${ S_\infty }$) is given by:

${ S_\infty = \frac{a}{1 - r} }$

provided that ${|r| < 1}$. This condition is essential because the sum to infinity converges only if the absolute value of the common ratio is less than 1. Now, let's apply these formulas to the given problem. We know that the fifteenth term (${ T_{15} }$) is ${ \frac{1}{8} }$ of the twelfth term (${ T_{12} }$). This can be written as:

${ T_{15} = \frac{1}{8} T_{12} }$

Using the formula for the nth term, we can express this relationship in terms of a and r:

${ ar^{14} = \frac{1}{8} ar^{11} }$

Since a cannot be zero (otherwise, the sum to infinity would be zero, contradicting the given information), we can divide both sides by a. Also, we can divide both sides by ${ r^{11} }$ (assuming r is not zero). This simplifies the equation to:

${ r^3 = \frac{1}{8} }$

Taking the cube root of both sides, we find:

${ r = \frac{1}{2} }$

Now that we have the common ratio, we can use the information about the sum to infinity to find the first term. We are given that ${ S_\infty = 5 }$. Using the formula for the sum to infinity:

${ 5 = \frac{a}{1 - \frac{1}{2}} }$

${ 5 = \frac{a}{\frac{1}{2}} }$

${ 5 = 2a }$

Dividing both sides by 2, we get:

${ a = \frac{5}{2} }$

Therefore, the first term of the geometric progression is ${ \frac{5}{2} }$, and the common ratio is ${ \frac{1}{2} }$. This completes the first part of the problem. We have successfully used the properties of geometric progressions to find the key parameters of the sequence. In the next section, we will tackle the second part of the problem, which involves finding the least number of terms needed for the sum to exceed a certain value.

(b) Finding the Least Number of Terms

In this part, we aim to find the least number of terms needed for the sum of the geometric progression to exceed a certain value. Let ${ S_n }$ denote the sum of the first n terms of the geometric progression. The formula for ${ S_n }$ is:

${ S_n = \frac{a(1 - r^n)}{1 - r} }$

We already know that the first term ${ a = \frac{5}{2} }$ and the common ratio ${ r = \frac{1}{2} }$. Substituting these values into the formula, we get:

${ S_n = \frac{\frac{5}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} }$

Simplifying the denominator, we have:

${ S_n = \frac{\frac{5}{2}(1 - (\frac{1}{2})^n)}{\frac{1}{2}} }$

${ S_n = 5(1 - (\frac{1}{2})^n) }$

We want to find the least integer n such that ${ S_n }$ exceeds a certain value. Let's consider the sum to be greater than 4.9. So, we need to find the smallest n such that:

${ 5(1 - (\frac{1}{2})^n) > 4.9 }$

Dividing both sides by 5, we get:

${ 1 - (\frac{1}{2})^n > \frac{4.9}{5} }$

${ 1 - (\frac{1}{2})^n > 0.98 }$

Rearranging the inequality, we have:

${ (\frac{1}{2})^n < 1 - 0.98 }$

${ (\frac{1}{2})^n < 0.02 }$

To solve for n, we can take the logarithm of both sides. It's convenient to use the natural logarithm (ln):

${ ln((\frac{1}{2})^n) < ln(0.02) }$

Using the property of logarithms that ${ ln(x^y) = y \cdot ln(x) }$, we get:

${ n \cdot ln(\frac{1}{2}) < ln(0.02) }$

Since ${ ln(\frac{1}{2}) }$ is negative, we need to reverse the inequality sign when dividing by it:

${ n > \frac{ln(0.02)}{ln(\frac{1}{2})} }$

Now, we calculate the values:

${ ln(0.02) \approx -3.912 }$

${ ln(\frac{1}{2}) \approx -0.693 }$

${ n > \frac{-3.912}{-0.693} }$

${ n > 5.645 }$

Since n must be an integer, the least integer value that satisfies this inequality is n = 6. Therefore, the least number of terms needed for the sum of the geometric progression to exceed 4.9 is 6.

Conclusion

In this article, we tackled a comprehensive problem involving geometric progressions. We successfully found the first term and common ratio of a geometric sequence given the relationship between its fifteenth and twelfth terms and its sum to infinity. We determined that the first term ( extit{a}) is ${ \frac{5}{2} }$ and the common ratio ( extit{r}) is ${ \frac{1}{2} }$. Additionally, we found the least number of terms needed for the sum of the geometric progression to exceed 4.9, which was determined to be 6 terms. This problem highlights the importance of understanding and applying the formulas for the nth term and the sum of geometric progressions. By breaking down the problem into manageable steps and utilizing algebraic manipulation and logarithmic properties, we were able to arrive at the solutions. The concepts explored here are crucial for anyone studying sequences and series in mathematics and have practical applications in various fields. Mastering these concepts provides a strong foundation for more advanced mathematical studies and real-world problem-solving scenarios. Through detailed explanations and step-by-step solutions, this article serves as a valuable resource for students and enthusiasts looking to deepen their understanding of geometric progressions.

This exercise underscores the significance of geometric progressions in both theoretical mathematics and practical applications. The skills developed in solving such problems—manipulating series formulas, working with logarithms, and interpreting inequalities—are broadly applicable across different areas of mathematics and science.