Expressing The Sum Of A Finite Geometric Series S₁₀₀ In Sigma Notation

Find the values of n, a, and r to express the sum of the finite geometric series $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$ in sigma notation.

In the realm of mathematics, geometric series hold a prominent position, particularly due to their frequent appearance in various applications ranging from finance to physics. A geometric series is characterized by a constant ratio between consecutive terms. Understanding how to represent and manipulate these series is crucial for problem-solving and deeper mathematical insights. In this article, we will delve into expressing the sum of a finite geometric series, specifically $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$, in sigma notation. Sigma notation, denoted by the Greek letter Σ, offers a concise and elegant way to represent the sum of a sequence of terms. It allows us to clearly define the terms being summed, the starting point of the summation, and the ending point. Our focus will be on identifying the key components of the geometric series, such as the common ratio and the number of terms, and then translating these into the appropriate sigma notation form. We will also explore the general formula for the sum of a finite geometric series and how it relates to the given expression for $S_{100}$. By the end of this discussion, you will have a solid understanding of how to convert a geometric series sum into sigma notation and appreciate the power and convenience of this notation in mathematical expressions.

Understanding Geometric Series

To fully appreciate the expression of a finite geometric series in sigma notation, it is crucial to first understand the fundamental concepts of geometric series. A geometric series is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This constant ratio, often denoted by 'r', is the defining characteristic of a geometric series. For instance, the sequence 2, 6, 18, 54, ... is a geometric series because each term is three times the previous term, making the common ratio 3. The first term of the series, denoted by 'a', is another essential component. In our example, the first term is 2.

The sum of a finite geometric series, where the series has a limited number of terms, can be calculated using a specific formula. This formula is particularly useful because it provides a direct way to compute the sum without having to add up each term individually. The formula for the sum of the first 'n' terms of a geometric series, denoted as $S_n$, is given by:

$S_n = a * \frac{1 - r^n}{1 - r}$, where:

• $S_n$ is the sum of the first 'n' terms,
• 'a' is the first term of the series,
• 'r' is the common ratio,
• 'n' is the number of terms.

This formula is valid when r ≠ 1. If r = 1, the series becomes a simple arithmetic series where each term is the same, and the sum is simply n * a. Now, let's consider the given expression: $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$. This expression represents the sum of a finite geometric series where we need to identify 'a', 'r', and 'n'. By comparing this expression with the general formula, we can deduce the values of these parameters. Notice that the numerator has a term $0.05^{100}$, which suggests that 0.05 is the common ratio raised to the power of the number of terms. The denominator (1 - 0.05) further confirms that 0.05 is indeed the common ratio. The entire expression closely resembles the formula for $S_n$, allowing us to extract the values needed for sigma notation.

Sigma Notation Explained

Sigma notation, also known as summation notation, is a powerful mathematical tool used to express the sum of a sequence of terms in a concise and efficient manner. The Greek capital letter Σ (sigma) is used to denote summation. Understanding sigma notation is essential for expressing and manipulating series, including geometric series, with ease. A typical sigma notation expression looks like this:

$\sum_{i=m}^{n} a_i$

Here's a breakdown of each component:

• Σ: The summation symbol, indicating that we are summing a series of terms.
• i: The index of summation, which is a variable that represents the term number in the series. It starts at a lower limit and increments by 1 until it reaches an upper limit.
• m: The lower limit of summation, which is the starting value of the index i. The summation begins with the m-th term.
• n: The upper limit of summation, which is the ending value of the index i. The summation ends with the n-th term.
• $a_i$: The summand, which is the expression that defines the terms being added. It is a function of the index i.

For example, the expression $\sum_{i=1}^{5} i^2$ represents the sum of the squares of the first five natural numbers. This means we would evaluate $1^2 + 2^2 + 3^2 + 4^2 + 5^2$. The index i starts at 1 and goes up to 5, and for each value of i, we compute $i^2$ and add it to the sum. Sigma notation is incredibly versatile and can be used to represent a wide variety of sums, including arithmetic series, geometric series, and more complex sequences. It provides a clear and unambiguous way to define the terms being summed, the range of the summation, and the pattern of the terms. This makes it an invaluable tool in mathematical analysis and various applications where summing sequences is necessary. In the context of geometric series, sigma notation allows us to express the sum in a compact form, highlighting the common ratio, the first term, and the number of terms being summed. Understanding the components of sigma notation is crucial for translating the sum of a finite geometric series, like our $S_{100}$, into this notation effectively.

Expressing S₁₀₀ in Sigma Notation

To express the given sum of the finite geometric series $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$ in sigma notation, we need to first identify the parameters of the series. Comparing the given expression with the formula for the sum of a finite geometric series, $S_n = a * \frac{1 - r^n}{1 - r}$, we can infer the values of 'a' (the first term), 'r' (the common ratio), and 'n' (the number of terms). In our case, the expression can be rewritten to resemble the formula more closely:

$S_{100} = 1 * \frac{1 - 0.05^{100}}{1 - 0.05}$

From this, we can deduce that:

• The first term, 'a', is 1.
• The common ratio, 'r', is 0.05.
• The number of terms, 'n', is 100.

Now that we have identified these parameters, we can express the sum in sigma notation. A geometric series can be represented in sigma notation as follows:

$\sum_{i=0}^{n-1} a * r^i$

Here, the index of summation, 'i', starts from 0 and goes up to n-1. Each term in the series is given by $a * r^i$, where 'a' is the first term and 'r' is the common ratio. For our specific series $S_{100}$, we have a = 1, r = 0.05, and n = 100. Plugging these values into the sigma notation, we get:

$\sum_{i=0}^{99} 1 * (0.05)^i$

This sigma notation represents the sum of the first 100 terms of a geometric series where the first term is 1 and the common ratio is 0.05. The index 'i' starts at 0, so the first term in the summation is $(0.05)^0 = 1$. The second term is $(0.05)^1 = 0.05$, the third term is $(0.05)^2 = 0.0025$, and so on, until the last term $(0.05)^{99}$. This notation concisely captures the sum of the geometric series up to 100 terms. Therefore, the sum $S_{100}$ can be expressed in sigma notation as $\sum_{i=0}^{99} (0.05)^i$. This representation not only simplifies the expression but also provides a clear understanding of the series being summed.

Variables in Sigma Notation for S₁₀₀

When expressing the sum of the finite geometric series $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$ in sigma notation, we have identified the key variables that define the series. These variables are crucial for accurately representing the sum in a compact and understandable form. As we determined earlier, the general form of a geometric series in sigma notation is:

$\sum_{i=0}^{n-1} a * r^i$

Where:

• 'n' represents the number of terms in the series.
• 'a' represents the first term of the series.
• 'r' represents the common ratio between consecutive terms.

In the context of $S_{100}$, we have already established the values for these variables. Let's reiterate and formalize these values for clarity:

• n: The number of terms in the series is 100. This is evident from the expression $S_{100}$, which implies that we are summing the first 100 terms of the geometric series. In the sigma notation, since the index 'i' starts from 0, the upper limit of the summation will be n - 1, which is 99 in this case.
• a: The first term of the series is 1. This can be inferred from the structure of the given sum. When we rewrite the sum as $1 * \frac{1 - 0.05^{100}}{1 - 0.05}$, it becomes clear that the first term is 1. In the sigma notation, this is the coefficient that multiplies the common ratio raised to the power of the index 'i'.
• r: The common ratio is 0.05. This is the constant factor by which each term is multiplied to get the next term in the series. It is the base of the exponent in the sigma notation expression. The common ratio is a defining characteristic of a geometric series and is crucial for both understanding and representing the series.

Therefore, in sigma notation, the variables for $S_{100}$ are:

• n = 100
• a = 1
• r = 0.05

These values are the foundation for expressing $S_{100}$ in sigma notation, allowing us to write the sum concisely as $\sum_{i=0}^{99} 1 * (0.05)^i$ or simply $\sum_{i=0}^{99} (0.05)^i$. Understanding and correctly identifying these variables is essential for translating any geometric series into sigma notation and vice versa.

Conclusion

In conclusion, expressing the sum of a finite geometric series in sigma notation is a valuable skill in mathematics. It allows for a concise and clear representation of the series, making it easier to analyze and manipulate. In this article, we focused on expressing $S_{100} = \frac{1 - 0.05^{100}}{1 - 0.05}$ in sigma notation by identifying the key variables: n (the number of terms), a (the first term), and r (the common ratio). We determined that for $S_{100}$, n = 100, a = 1, and r = 0.05. This allowed us to write the sum in sigma notation as $\sum_{i=0}^{99} (0.05)^i$. Understanding the components of both geometric series and sigma notation is crucial for this process. The formula for the sum of a finite geometric series, $S_n = a * \frac{1 - r^n}{1 - r}$, provides a direct link to the parameters needed for sigma notation. Sigma notation, with its summation symbol Σ, index of summation, limits, and summand, offers a powerful way to represent sums in a compact form. By mastering these concepts, one can effectively translate between explicit sums and sigma notation, enhancing their mathematical toolkit. The ability to express sums in sigma notation is not only useful in theoretical mathematics but also in various practical applications, such as in finance for calculating annuities or in physics for modeling systems with exponential decay. Therefore, a solid understanding of these concepts is essential for anyone working with mathematical series and sequences.