Geometric Series Summation Explained ∑n=14(-2)(-3)n-1

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In the realm of mathematics, geometric series hold a significant position, particularly in areas like calculus, financial mathematics, and physics. Understanding how to calculate the sum of a geometric series is crucial for solving a wide range of problems. This article delves into the intricacies of geometric series, providing a comprehensive explanation of how to determine their sums, with a focus on the specific example: ∑n=14(-2)(-3)n-1. We will explore the fundamental concepts, formulas, and steps involved in solving this type of problem, ensuring a clear and thorough understanding.

Defining Geometric Series

To understand how to calculate the sum of a geometric series, we must first define what it is. A geometric series is a sequence of numbers where each term is multiplied by a constant factor to obtain the next term. This constant factor is known as the common ratio (r). The general form of a geometric series is:

a, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio. The sum of a geometric series can be finite (if we are summing a specific number of terms) or infinite (if the series goes on indefinitely). Our focus here is on finite geometric series, specifically the sum of the first four terms of the series defined by ∑n=14(-2)(-3)n-1.

The Formula for the Sum of a Finite Geometric Series

The formula for the sum of the first 'n' terms of a geometric series is a powerful tool for quickly finding the sum without having to add each term individually. The formula is given by:

S_n = a(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 to be summed

This formula is applicable as long as the common ratio 'r' is not equal to 1. If r = 1, the series becomes a simple arithmetic series, and the sum can be calculated differently. However, for most geometric series problems, this formula provides a straightforward method for finding the sum.

Applying the Formula to the Given Series

Now, let's apply this formula to the specific geometric series given: ∑n=14(-2)(-3)n-1. This notation represents the sum of the first four terms of a geometric series where the nth term is given by (-2)(-3)n-1. To use the formula, we need to identify 'a', 'r', and 'n'.

Identifying 'a', 'r', and 'n'

  1. Identifying the First Term (a): The first term (a) is the term when n = 1. Plugging n = 1 into the expression (-2)(-3)n-1 gives us:

    a = (-2)(-3)1-1 = (-2)(-3)0 = (-2)(1) = -2

    So, the first term of the series is -2.

  2. Identifying the Common Ratio (r): The common ratio (r) is the factor by which each term is multiplied to get the next term. To find 'r', we can calculate the second term and divide it by the first term. The second term is when n = 2:

    Term 2 = (-2)(-3)2-1 = (-2)(-3)1 = 6

    So, the common ratio r is:

    r = Term 2 / Term 1 = 6 / -2 = -3

    Therefore, the common ratio is -3.

  3. Identifying the Number of Terms (n): The summation notation ∑n=14 indicates that we are summing the series from n = 1 to n = 4. This means there are 4 terms in the series. So, n = 4.

Plugging the Values into the Formula

Now that we have identified a = -2, r = -3, and n = 4, we can plug these values into the formula for the sum of a finite geometric series:

S_n = a(1 - r^n) / (1 - r)

S_4 = -2(1 - (-3)^4) / (1 - (-3))

Let's break down the calculation step by step:

  1. Calculate (-3)^4:

    (-3)^4 = (-3) * (-3) * (-3) * (-3) = 81

  2. Substitute this value back into the equation:

    S_4 = -2(1 - 81) / (1 - (-3))

  3. Simplify the terms inside the parentheses:

    S_4 = -2(-80) / (1 + 3)

  4. Continue simplifying:

    S_4 = 160 / 4

  5. Finally, divide to get the sum:

    S_4 = 40

Thus, the sum of the first four terms of the geometric series ∑n=14(-2)(-3)n-1 is 40.

Alternative Method: Direct Calculation

While the formula provides an efficient way to calculate the sum, it's also helpful to understand how to find the sum by directly calculating and adding the terms. This method can be particularly useful for smaller series where the number of terms is manageable.

Calculating Each Term

To find the sum directly, we calculate each term of the series and then add them together.

  1. Term 1 (n = 1):

    (-2)(-3)1-1 = (-2)(-3)0 = (-2)(1) = -2

  2. Term 2 (n = 2):

    (-2)(-3)2-1 = (-2)(-3)1 = (-2)(-3) = 6

  3. Term 3 (n = 3):

    (-2)(-3)3-1 = (-2)(-3)2 = (-2)(9) = -18

  4. Term 4 (n = 4):

    (-2)(-3)4-1 = (-2)(-3)3 = (-2)(-27) = 54

Summing the Terms

Now, we add the terms together:

Sum = -2 + 6 + (-18) + 54

Sum = 4 + (-18) + 54

Sum = -14 + 54

Sum = 40

As we can see, the sum calculated by directly adding the terms is also 40, which matches the result obtained using the formula. This method provides a clear, step-by-step way to understand how the sum is derived.

Common Mistakes and How to Avoid Them

When working with geometric series, it's easy to make mistakes if certain details are overlooked. Here are some common mistakes and how to avoid them:

  1. Incorrectly Identifying the First Term (a): Ensure that you correctly substitute n = 1 into the expression to find the first term. Sometimes, the series may not start at n = 1, so pay close attention to the lower limit of the summation.

  2. Miscalculating the Common Ratio (r): The common ratio is the factor by which each term is multiplied. Make sure to divide a term by its preceding term to find 'r'. Avoid dividing by the term number (n) or making sign errors.

  3. Using the Wrong Number of Terms (n): Carefully consider the upper and lower limits of the summation to determine the number of terms. For example, ∑n=37 would have 7 - 3 + 1 = 5 terms, not just 7 - 3 = 4 terms.

  4. Applying the Formula Incorrectly: Double-check that you have substituted the values of 'a', 'r', and 'n' correctly into the formula. A small mistake in substitution can lead to a wrong answer. Also, remember that the formula S_n = a(1 - r^n) / (1 - r) is valid only when r ≠ 1.

  5. Arithmetic Errors: Be cautious with arithmetic operations, especially when dealing with negative numbers and exponents. Double-check each step to minimize the chances of making a mistake.

By being mindful of these common pitfalls, you can significantly improve your accuracy when solving geometric series problems.

Conclusion

In summary, finding the sum of a geometric series involves understanding the fundamental concepts, correctly identifying the first term (a), the common ratio (r), and the number of terms (n), and applying the appropriate formula. For the given series, ∑n=14(-2)(-3)n-1, we identified a = -2, r = -3, and n = 4. Using the formula S_n = a(1 - r^n) / (1 - r), we calculated the sum to be 40. We also verified this result by directly calculating and adding the terms, reinforcing the understanding of the summation process.

Geometric series are a vital concept in mathematics, with applications in various fields. Mastering the techniques for calculating their sums equips you with a powerful tool for problem-solving and analytical thinking. By paying attention to details, avoiding common mistakes, and practicing regularly, you can confidently tackle geometric series problems and enhance your mathematical skills.

This comprehensive guide has provided a detailed explanation of how to calculate the sum of a geometric series, with a specific focus on the example ∑n=14(-2)(-3)n-1. Whether you are a student learning this concept for the first time or someone looking to refresh your knowledge, this article serves as a valuable resource for understanding and applying the principles of geometric series summation.

The correct answer is C. 40.