Determining The Formula For S_n In The Sequence A_n = 3(0.7)^n

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In the realm of mathematics, sequences and series play a pivotal role in understanding patterns and predicting future values. One common type of sequence is the geometric sequence, where each term is obtained by multiplying the previous term by a constant factor. To delve deeper into this concept, let's explore the sequence defined by a_n = 3(0.7)^n and unravel the rule that governs its partial sums, denoted as S_n. We will analyze the given options and derive the correct formula for S_n, providing a comprehensive understanding of the underlying principles. This exploration will not only solidify your understanding of geometric sequences and series but also enhance your problem-solving skills in mathematics.

Understanding Geometric Sequences and Series

Before we dive into the specifics of the problem, let's establish a solid foundation by revisiting the concepts of geometric sequences and series. A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant, known as the common ratio (r). The general form of a geometric sequence is: a, ar, ar^2, ar^3, ... where 'a' represents the first term.

A geometric series, on the other hand, is the sum of the terms in a geometric sequence. The sum of the first 'n' terms of a geometric series, denoted as S_n, can be calculated using the following formula:

S_n = a(1 - r^n) / (1 - r), where r ≠ 1

This formula is crucial for solving problems involving geometric series, as it allows us to efficiently calculate the sum of a large number of terms without having to manually add them up. Understanding the derivation and application of this formula is essential for mastering geometric sequences and series.

Delving into the Given Sequence: a_n = 3(0.7)^n

Now, let's focus on the sequence provided in the problem: a_n = 3(0.7)^n. This sequence is a geometric sequence because each term is obtained by multiplying the previous term by a constant factor, which in this case is 0.7. To fully grasp the characteristics of this sequence, let's identify the first term (a) and the common ratio (r).

By substituting n = 1 into the formula, we find the first term:

a_1 = 3(0.7)^1 = 3(0.7) = 2.1

Therefore, the first term of the sequence is 2.1. The common ratio (r) is the constant factor that multiplies each term to get the next term, which is 0.7 in this case. With the first term and common ratio identified, we can now proceed to determine the rule that defines S_n, the sum of the first 'n' terms of this sequence.

Determining the Rule for S_n

To find the rule for S_n, we will utilize the formula for the sum of the first 'n' terms of a geometric series:

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

We already know that the first term (a) is 2.1 and the common ratio (r) is 0.7. Substituting these values into the formula, we get:

S_n = 2.1(1 - (0.7)^n) / (1 - 0.7)

Simplifying the denominator, we have:

S_n = 2.1(1 - (0.7)^n) / 0.3

Now, we can divide 2.1 by 0.3:

S_n = 7(1 - (0.7)^n)

Therefore, the rule that defines S_n for the sequence a_n = 3(0.7)^n is S_n = 7(1 - (0.7)^n). This result aligns with option C in the given choices.

Analyzing the Answer Choices

Now that we have derived the correct formula for S_n, let's examine the given answer choices to solidify our understanding and eliminate any potential confusion.

  • A. S_n = 7(0.7)^n: This option is incorrect because it does not account for the summation of terms in the series. It only represents a scaled version of the nth term of the sequence.
  • B. S_n = 2.1(0.7)^n: Similar to option A, this choice fails to represent the sum of the terms. It simply scales the nth term by the first term of the sequence.
  • C. S_n = 7(1 - (0.7)^n): This is the correct answer, as we derived earlier using the formula for the sum of a geometric series. It accurately represents the sum of the first 'n' terms of the sequence.
  • D. S_n = 2.1(1 - (0.7)^n): This option is incorrect because it uses the first term (2.1) as the scaling factor instead of the correct value of 7, which we obtained by dividing 2.1 by (1 - 0.7).

By carefully analyzing each option and comparing it to our derived formula, we can confidently conclude that option C is the correct rule that defines S_n for the given sequence.

Key Takeaways and Applications

This problem provides valuable insights into geometric sequences and series, highlighting the importance of understanding the underlying formulas and applying them correctly. Let's summarize the key takeaways:

  • A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant common ratio.
  • A geometric series is the sum of the terms in a geometric sequence.
  • The sum of the first 'n' terms of a geometric series can be calculated using the formula: S_n = a(1 - r^n) / (1 - r), where 'a' is the first term and 'r' is the common ratio (r ≠ 1).
  • Identifying the first term and common ratio is crucial for solving problems involving geometric sequences and series.

The concepts explored in this problem have wide-ranging applications in various fields, including:

  • Finance: Calculating compound interest, loan payments, and annuities.
  • Physics: Modeling radioactive decay and oscillations.
  • Computer Science: Analyzing algorithms and data structures.
  • Economics: Predicting economic growth and inflation.

By mastering the principles of geometric sequences and series, you can unlock a powerful toolset for solving problems in diverse domains.

Practice Problems to Enhance Your Understanding

To further solidify your understanding of geometric sequences and series, let's explore some practice problems. These problems will challenge you to apply the concepts we've discussed in different scenarios, helping you develop your problem-solving skills.

Problem 1:

Consider the geometric sequence: 2, 6, 18, 54, ...

  • Find the common ratio (r).
  • Determine the 8th term of the sequence.
  • Calculate the sum of the first 6 terms (S_6).

Problem 2:

The sum of the first 5 terms of a geometric series is 93. The first term is 3. Find the common ratio and the fifth term.

Problem 3:

A ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces to 3/4 of its previous height. What is the total distance the ball travels before it comes to rest?

These practice problems cover various aspects of geometric sequences and series, including finding the common ratio, determining specific terms, calculating sums, and applying the concepts to real-world scenarios. By working through these problems, you'll gain a deeper understanding of the material and improve your ability to solve similar problems in the future.

Conclusion

In conclusion, we successfully identified the rule that defines S_n for the sequence a_n = 3(0.7)^n as S_n = 7(1 - (0.7)^n). This was achieved by understanding the fundamental concepts of geometric sequences and series, applying the formula for the sum of a geometric series, and carefully analyzing the given answer choices. The exploration of geometric sequences and series provides a valuable foundation for understanding patterns, predicting future values, and solving problems in various fields. By mastering these concepts and practicing problem-solving techniques, you can enhance your mathematical skills and unlock a world of possibilities.

By delving into the intricacies of geometric sequences and series, we've not only answered the specific question but also gained a broader understanding of mathematical principles that extend far beyond this single problem. The ability to recognize patterns, apply formulas, and analyze solutions is crucial for success in mathematics and many other disciplines. As you continue your mathematical journey, remember to build upon the knowledge you've gained and embrace the challenges that come your way. With perseverance and a solid foundation, you can achieve your mathematical goals and unlock your full potential.