Video Masalah "Barisan Geometri" |Orientasi Masalah | PBL

Introduction

This tutorial focuses on understanding geometric sequences, a key concept in mathematics that deals with sequences where each term is found by multiplying the previous term by a constant factor. This guide will help you grasp the fundamentals of geometric sequences and apply problem-based learning (PBL) strategies to solve related problems effectively.

Step 1: Understand the Definition of Geometric Sequences

• A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
• Example: In the sequence 2, 6, 18, 54, the common ratio is 3 (6/2 = 3, 18/6 = 3, etc.).

Step 2: Identify the Common Ratio

• To identify the common ratio:
  • Divide any term by its preceding term.
  • Ensure the result is consistent across the sequence.
• Example Calculation:
  • For the sequence 4, 12, 36:
    • 12/4 = 3
    • 36/12 = 3
  • Thus, the common ratio is 3.

Step 3: Formulate the General Term

• The nth term of a geometric sequence can be represented with the formula:
  • ( a_n = a_1 \cdot r^{(n-1)} )
  • Where:
    • ( a_n ) = nth term
    • ( a_1 ) = first term
    • ( r ) = common ratio
    • ( n ) = term number
• Example:
  • For a sequence with first term 5 and common ratio 2, the 4th term is calculated as:
  • ( a_4 = 5 \cdot 2^{(4-1)} = 5 \cdot 8 = 40 )

Step 4: Solve Problems Using PBL Approach

• Apply the PBL approach by:
  • Identifying real-world scenarios where geometric sequences apply, such as finance (compound interest) or biology (population growth).
  • Formulating problems based on these scenarios.
• Example Problem:
  • If a bacteria population doubles every hour and starts with 100 bacteria, how many will there be after 5 hours?
  • Solution:
    • Use the formula with ( a_1 = 100 ) and ( r = 2 ):
    • ( a_6 = 100 \cdot 2^{(5)} = 100 \cdot 32 = 3200 )

Step 5: Common Pitfalls to Avoid

• Confusing geometric sequences with arithmetic sequences, where terms are added by a constant.
• Miscalculating the common ratio.
• Forgetting to adjust for the position of the term in the sequence when applying the general term formula.

Conclusion

Understanding geometric sequences is essential for tackling various mathematical problems, especially those involving exponential growth. By mastering the definition, identifying the common ratio, and applying the general term formula, you can effectively solve problems using the PBL approach. Practice with real-life scenarios to deepen your understanding, and always double-check your calculations to avoid common pitfalls.