Barisan dan Deret Bagian 4 - Deret Geometri Matematika Wajib Kelas 11

Introduction

This tutorial will guide you through the concept of geometric series, specifically focusing on how to find the sum of a geometric series. This topic is essential for 11th-grade mathematics and is integral for understanding more advanced mathematical concepts. By the end of this guide, you will have a clear understanding of how to calculate the sum of a geometric series through examples and practical applications.

Step 1: Understanding Geometric Series

• A geometric series is formed by adding the terms of a geometric sequence.

• A geometric sequence has a constant ratio between consecutive terms, known as the common ratio (r).

• The general form of a geometric series can be represented as:

  S_n = a + ar + ar^2 + ... + ar^(n-1)

  where:

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

Step 2: Finding the Sum of a Finite Geometric Series

• The formula to calculate the sum of the first n terms of a geometric series is:

  S_n = a * (1 - r^n) / (1 - r) (when r ≠ 1)

• To use this formula effectively:

  1. Identify the first term (a).
  2. Determine the common ratio (r).
  3. Decide how many terms (n) you want to add together.

Practical Tip

• If the common ratio (r) is greater than 1, the series will grow quickly. If it's between 0 and 1, the series will converge towards a limit.

Step 3: Example Calculation

Let’s go through a specific example to clarify the process:

Example 1

• Given:

  • First term (a) = 3
  • Common ratio (r) = 2
  • Number of terms (n) = 5

• To find S_n:

  1. Apply the formula:

     S_5 = 3 * (1 - 2^5) / (1 - 2)

  2. Calculate:

     S_5 = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 93

Example 2

• Given:

  • First term (a) = 4
  • Common ratio (r) = 0.5
  • Number of terms (n) = 6

• To find S_n:

  1. Apply the formula:

     S_6 = 4 * (1 - 0.5^6) / (1 - 0.5)

  2. Calculate:

     S_6 = 4 * (1 - 0.015625) / 0.5 = 4 * 0.984375 / 0.5 = 7.875

Step 4: Additional Examples

• Continue practicing with more examples to solidify your understanding. Here are some pointers for examples:
  1. Choose different first terms and common ratios.
  2. Vary the number of terms to see how it affects the sum.

Conclusion

In this tutorial, you learned how to calculate the sum of a geometric series using the formula S_n = a * (1 - r^n) / (1 - r). We discussed the importance of identifying the first term, common ratio, and the number of terms. By practicing with the examples provided, you can gain confidence in solving problems related to geometric series. For further learning, consider exploring more complex series or diving into infinite geometric series for advanced understanding.