Baris dan Deret Geometri | Matematika Kelas X Fase E Kurikulum Merdeka

Introduction

This tutorial will guide you through the concepts of geometric sequences and series, which are fundamental topics in mathematics, particularly in the context of the Kurikulum Merdeka for Class X. Understanding these concepts will enhance your problem-solving skills and prepare you for more advanced mathematical studies.

Step 1: Understanding Geometric Sequences

• Definition: A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio.

• Formula: The n-th term of a geometric sequence can be expressed as:

  [ a_n = a_1 \times r^{(n-1)} ]

  Where:

  • (a_n) is the n-th term,
  • (a_1) is the first term,
  • (r) is the common ratio,
  • (n) is the term number.

• Example: If the first term (a_1) is 3 and the common ratio (r) is 2, the sequence is:

  • 3, 6, 12, 24, ...

Step 2: Recognizing the Common Ratio

• To find the common ratio (r), divide any term by the previous term.
• Example: In the sequence 4, 8, 16:
  • (r = \frac{8}{4} = 2)
  • (r = \frac{16}{8} = 2)

Step 3: Exploring Geometric Series

• Definition: A geometric series is the sum of the terms of a geometric sequence.

• Formula: The sum (S_n) of the first n terms of a geometric series can be calculated using the formula:

  [ S_n = a_1 \times \frac{1 - r^n}{1 - r} \quad \text{(if } r \neq 1\text{)} ]

• Example: For the first three terms of a geometric sequence where (a_1 = 2) and (r = 3):

  • (S_3 = 2 \times \frac{1 - 3^3}{1 - 3} = 2 \times \frac{1 - 27}{-2} = 26)

Step 4: Solving Example Problems

• Problem 1: Find the 5th term of a sequence where (a_1 = 5) and (r = 3).

  • Solution:
    • Use the formula: (a_5 = 5 \times 3^{(5-1)} = 5 \times 81 = 405)

• Problem 2: Calculate the sum of the first 4 terms of the sequence where (a_1 = 1) and (r = 2).

  • Solution:
    • (S_4 = 1 \times \frac{1 - 2^4}{1 - 2} = 1 \times \frac{1 - 16}{-1} = 15)

Step 5: Practical Applications

• Geometric sequences and series are used in various fields such as finance (compound interest), computer science (algorithm analysis), and physics (exponential decay).

Conclusion

Understanding geometric sequences and series is crucial for solving a variety of mathematical problems. Remember to identify the first term and the common ratio to apply the formulas effectively. Practice solving problems to reinforce these concepts, and explore their applications in real-world scenarios. For further study, consider diving into more complex mathematical series or engaging in problem-solving exercises related to geometric sequences.