Calculating The Sum Of A Finite Geometric Series A Step By Step Guide

In the realm of mathematics, sequences and series play a pivotal role in various applications, from financial modeling to physics. Among these, geometric series hold a special place due to their unique properties and predictable behavior. A geometric series is essentially the sum of terms in a geometric sequence, where each term is multiplied by a constant factor, known as the common ratio, to obtain the next term. Calculating the sum of a finite geometric series is a fundamental task, and a well-established formula provides an efficient way to achieve this. This formula not only simplifies the process but also offers valuable insights into the convergence and divergence of infinite geometric series. Mastering this concept is crucial for anyone delving deeper into mathematical analysis and its applications.

The geometric series formula provides a direct method for finding the sum without having to manually add each term, especially when dealing with a large number of terms. This formula is represented as:

S_n = a rac{1 - r^n}{1 - r}

Where:

• $S_n$ is the sum of the first n terms of the series.
• a is the first term of the series.
• r is the common ratio (the factor by which each term is multiplied to get the next term).
• n is the number of terms in the series.

This formula is derived from algebraic manipulation of the series and provides an elegant way to calculate the sum. However, it is crucial to understand the conditions under which this formula is valid. Notably, the formula is applicable when the common ratio r is not equal to 1. When r is 1, the series becomes a simple arithmetic series, and the sum can be calculated using a different formula. Furthermore, the formula is particularly useful for finite geometric series, where the number of terms n is a finite number. For infinite geometric series, the sum converges to a finite value only when the absolute value of r is less than 1; otherwise, the series diverges.

Understanding the geometric series formula also provides insights into the behavior of sequences and series in general. For instance, the formula reveals how the sum of the series depends on the initial term a, the common ratio r, and the number of terms n. A small change in any of these parameters can significantly affect the sum of the series. Moreover, the formula is closely related to the concept of convergence and divergence of infinite series, which is a cornerstone of mathematical analysis. By analyzing the formula, we can determine the conditions under which an infinite geometric series converges to a finite value, which is crucial for applications in areas such as calculus and differential equations.

Now, let's apply this knowledge to the specific problem at hand. We are given the geometric series expressed in sigma notation:

$$\sum_{i=1}^7 2\left(\frac{4}{7}\right)^{i-1}$$

Our objective is to find the sum of this finite geometric series using the formula for $S_n$. To do this, we first need to identify the key parameters of the series: the first term (a), the common ratio (r), and the number of terms (n). Understanding these parameters is crucial for correctly applying the formula and obtaining the correct result. A careful analysis of the series notation will reveal these values and set the stage for a straightforward calculation.

By examining the sigma notation, we can deduce the following:

• The first term (a) can be found by substituting i = 1 into the expression $2\left(\frac{4}{7}\right)^{i-1}$. This gives us:

  $$a = 2\left(\frac{4}{7}\right)^{1-1} = 2\left(\frac{4}{7}\right)^0 = 2 \times 1 = 2$$

  Thus, the first term of the series is 2. This value serves as the starting point for the geometric progression and is a critical component in determining the overall sum of the series. The initial term significantly influences the magnitude of the sum, especially when combined with the common ratio and the number of terms. A larger initial term generally leads to a larger sum, provided the common ratio is not significantly less than 1. In the context of real-world applications, the initial term might represent the starting value of an investment, the initial population size, or any other quantity that grows or decays geometrically.

• The common ratio (r) is the factor by which each term is multiplied to obtain the next term. In this case, it is evident from the expression that the common ratio is $\frac{4}{7}$. This means that each term in the series is $\frac{4}{7}$ times the previous term. The common ratio plays a pivotal role in determining the behavior of the series. If the absolute value of the common ratio is less than 1, the series converges, meaning that the sum of the series approaches a finite value as the number of terms increases. Conversely, if the absolute value of the common ratio is greater than or equal to 1, the series diverges, and the sum grows without bound. In practical terms, a common ratio less than 1 might represent a decay process, such as radioactive decay or the depreciation of an asset, while a common ratio greater than 1 might represent a growth process, such as compound interest or population growth.

• The number of terms (n) is determined by the limits of the summation. Since i ranges from 1 to 7, there are 7 terms in the series. This indicates that we are dealing with a finite geometric series, which has a definite sum that can be calculated using the formula. The number of terms is a crucial factor in determining the sum of the series, especially when the common ratio is close to 1. In such cases, the sum can increase significantly as the number of terms increases. In practical applications, the number of terms might represent the duration of an investment, the number of generations in a population, or any other discrete time period over which a geometric process occurs.

With these parameters identified, we are now ready to substitute them into the formula for the sum of a finite geometric series.

Now that we have identified the first term (a = 2), the common ratio (r = $\frac{4}{7}$), and the number of terms (n = 7), we can substitute these values into the formula for the sum of a finite geometric series:

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

Plugging in the values, we get:

$$S_7 = 2 \frac{1 - \left(\frac{4}{7}\right)^7}{1 - \frac{4}{7}}$$

This expression represents the sum of the first 7 terms of the geometric series. To calculate the sum, we need to simplify this expression by performing the arithmetic operations. This involves evaluating the power, subtracting the fractions, and then multiplying the result. The order of operations is crucial to ensure that we arrive at the correct answer. A common mistake is to perform the subtraction in the numerator or denominator before evaluating the power, which can lead to an incorrect result. Therefore, it is essential to follow the correct order of operations to avoid errors.

First, let's calculate $\left(\frac{4}{7}\right)^7$:

$$\left(\frac{4}{7}\right)^7 = \frac{4^7}{7^7} = \frac{16384}{823543}$$

This step involves raising both the numerator and the denominator to the power of 7. The resulting fraction represents the seventh power of the common ratio. The value of this term is crucial in determining the overall sum of the series, especially when the number of terms is large. As the common ratio is less than 1, its powers decrease rapidly, and this term contributes less and less to the sum as n increases. In the context of infinite geometric series, this behavior is key to understanding convergence, as the powers of the common ratio approach zero, and the sum converges to a finite value.

Next, we subtract this from 1:

$$1 - \frac{16384}{823543} = \frac{823543 - 16384}{823543} = \frac{807159}{823543}$$

This step calculates the numerator of the fraction in the formula. The subtraction gives us the difference between 1 and the seventh power of the common ratio. This difference represents the factor by which the initial term is multiplied to obtain the sum of the series. The magnitude of this difference depends on the common ratio and the number of terms. When the common ratio is close to 1, this difference can be small, and the sum of the series can be significantly larger than the initial term. Conversely, when the common ratio is close to 0, this difference approaches 1, and the sum of the series is close to the initial term.

Now, let's simplify the denominator:

$$1 - \frac{4}{7} = \frac{7 - 4}{7} = \frac{3}{7}$$

This step calculates the denominator of the fraction in the formula. The subtraction gives us the difference between 1 and the common ratio. This difference is a crucial factor in determining the sum of the series. When the common ratio is close to 1, this difference is small, and the sum of the series can be very large. Conversely, when the common ratio is close to 0, this difference is close to 1, and the sum of the series is close to the product of the initial term and the number of terms.

Substituting these results back into the formula, we get:

$$S_7 = 2 \frac{\frac{807159}{823543}}{\frac{3}{7}} = 2 \times \frac{807159}{823543} \times \frac{7}{3}$$

This expression represents the sum of the series in terms of fractions. To simplify this further, we can perform the multiplication and reduce the fraction to its simplest form. This step involves multiplying the numerators and the denominators and then finding the greatest common divisor (GCD) to simplify the fraction. Simplifying the fraction makes the result easier to understand and compare with other values. In some cases, it might be necessary to convert the fraction to a decimal to get a better sense of its magnitude.

$$S_7 = 2 \times \frac{807159 \times 7}{823543 \times 3} = 2 \times \frac{5649993}{2470629} = \frac{11299986}{2470629}$$

Now, we simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:

$$S_7 = \frac{11299986 \div 3}{2470629 \div 3} = \frac{3766662}{823543}$$

Thus, the sum of the finite geometric series is $\frac{3766662}{823543}$. This fraction represents the exact sum of the first 7 terms of the series. While it is a precise answer, it might be more informative to express it as a mixed number or a decimal. Converting the fraction to a decimal gives us an approximate value that can be easier to interpret. In many practical applications, an approximate value is sufficient, and it can be more useful than the exact fraction.

Therefore, the sum of the finite geometric series $\sum_{i=1}^7 2\left(\frac{4}{7}\right)^{i-1}$ is $\frac{3766662}{823543}$.

In summary, we have successfully found the sum of the finite geometric series by applying the formula for $S_n$. The process involved identifying the first term, common ratio, and number of terms, and then substituting these values into the formula. We then simplified the resulting expression to obtain the final answer. This problem demonstrates the power and elegance of mathematical formulas in solving complex problems. The geometric series formula provides a straightforward method for calculating the sum of a series, which would be tedious and time-consuming to compute manually, especially for a large number of terms. The ability to apply such formulas is a crucial skill in mathematics and its applications.

Understanding geometric series and their sums is fundamental in various fields, including finance, physics, and computer science. For instance, in finance, geometric series are used to model compound interest and annuities. In physics, they appear in the analysis of oscillations and waves. In computer science, they are used in the analysis of algorithms and data structures. The concepts and techniques discussed in this article provide a solid foundation for further exploration of these topics. By mastering the geometric series formula and its applications, students and professionals can tackle a wide range of problems and gain a deeper appreciation for the power of mathematics.