Pola Bilangan (5) | Barisan dan Deret Geometri

Introduction

This tutorial focuses on the concept of geometric sequences and series, an important topic in mathematics, particularly in the eighth-grade curriculum. Understanding these concepts is essential for solving various mathematical problems and can be applied in real-world scenarios such as finance and computer science.

Step 1: Understand Geometric Sequences

A geometric sequence is a sequence 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.

Key Points

• Common Ratio: The factor by which we multiply each term to get the next.
  • Example: In the sequence 2, 4, 8, 16, the common ratio is 2.
• General Formula: The n-th term of a geometric sequence can be expressed as:
  • ( a_n = a_1 \cdot r^{(n-1)} )
  • Where ( a_n ) is the n-th term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.

Step 2: Identify Geometric Series

A geometric series is the sum of the terms of a geometric sequence.

Key Points

• Finite Geometric Series: The sum of a finite number of terms can be calculated using the formula:
  • ( S_n = a_1 \frac{(1 - r^n)}{(1 - r)} ) for ( r \neq 1 )
• Infinite Geometric Series: If the absolute value of the common ratio is less than 1, the series can be summed to infinity:
  • ( S = \frac{a_1}{(1 - r)} )

Step 3: Solve Examples

To solidify your understanding, try solving some examples:

Example 1: Find the 5th term of the sequence

1. Given the sequence 3, 6, 12, 24 (first term ( a_1 = 3 ), common ratio ( r = 2 )).
2. Use the formula ( a_n = a_1 \cdot r^{(n-1)} ):
   • ( a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 16 = 48 )

Example 2: Calculate the sum of the first 4 terms

1. Using the same sequence, apply the finite geometric series formula:
   • ( S_n = 3 \frac{(1 - 2^4)}{(1 - 2)} = 3 \frac{(1 - 16)}{(-1)} = 3 \cdot 15 = 45 )

Step 4: Practice Problems

To enhance your skills, try these practice problems:

1. Find the 6th term of the sequence 5, 10, 20, 40.
2. Calculate the sum of the first 5 terms of the sequence 1, 3, 9, 27.

Conclusion

In this tutorial, we explored the fundamentals of geometric sequences and series, including their definitions, formulas, and applications. Understanding these concepts will greatly enhance your mathematical skills, particularly in areas like finance and data analysis. Continue practicing with various examples to solidify your knowledge, and consider exploring related topics such as arithmetic sequences for a broader understanding of number patterns.