(Part 4/4) BARISAN DAN DERET MATEMATIKA KELAS 10 KURIKULUM MERDEKA #kurikulummerdeka #deretgeometri

Introduction

This tutorial covers the concept of infinite geometric series, a fundamental topic in mathematics, especially for 10th-grade students following the Merdeka Curriculum. You will learn about the notation, formula, and proofs related to infinite geometric series, enabling you to understand how to calculate the sum of an infinite series.

Step 1: Understanding Infinite Geometric Series

• An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely.
• It is commonly denoted as S∞.
• The series can be represented as:
  • S∞ = a + ar + ar² + ar³ + ...
  • Where:
    • a = first term
    • r = common ratio (the factor by which we multiply each term to get the next)

Step 2: Identifying Conditions for Convergence

• An infinite geometric series converges (has a finite sum) if the absolute value of the common ratio is less than 1 (|r| < 1).
• If |r| ≥ 1, the series diverges (the sum approaches infinity).

Step 3: The Formula for the Sum of an Infinite Geometric Series

• The sum of an infinite geometric series can be calculated using the formula:

  S∞ = a / (1 - r)
  

• Key points:

  • Ensure |r| < 1 for the series to converge.
  • This formula allows you to find the sum without needing to add an infinite number of terms manually.

Step 4: Example Calculation

1. Identify the first term (a) and common ratio (r):

   • Example: a = 2, r = 1/2.

2. Check the common ratio condition:

   • Here, |1/2| < 1, so the series converges.

3. Use the formula to find S∞:

   S∞ = 2 / (1 - 1/2)
   S∞ = 2 / (1/2)
   S∞ = 4
   

Step 5: Practical Applications

• Infinite geometric series are used in various fields, including:
  • Calculating the total distance traveled in physics when the distance halves each time.
  • Financial calculations such as present value of annuities.

Conclusion

Understanding infinite geometric series is crucial for solving problems in mathematics and real-world applications. By mastering the formula and the conditions for convergence, you can tackle various mathematical challenges. Practice with different values of a and r to reinforce your understanding, and explore applications in finance and physics for a broader perspective on the topic.