Matematika SMA - Barisan dan Deret (1) - Barisan Aritmatika, Rumus Barisan Aritmatika (A)

Introduction

This tutorial will guide you through the concepts of arithmetic sequences and series, which are crucial topics in high school mathematics. Understanding these concepts will enhance your problem-solving skills in algebra and prepare you for more advanced mathematical studies.

Step 1: Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference."

Key Points

• Common Difference (d): The fixed amount added to each term to get to the next term.
• General Form: The nth term of an arithmetic sequence can be expressed as:

  a_n = a_1 + (n - 1) * d
  

  Where

  • a_n is the nth term
  • a_1 is the first term
  • d is the common difference
  • n is the term number

Practical Advice

• To identify an arithmetic sequence, look for a consistent difference between terms. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

Step 2: Finding the Common Difference

To find the common difference, subtract the first term from the second term.

Example

Given the sequence: 4, 7, 10, 13

• Common difference (d) = 7 - 4 = 3

Step 3: Calculating Specific Terms

Use the formula for the nth term to calculate any term in the sequence.

Example Calculation

Find the 5th term of the sequence where the first term is 2 and the common difference is 3.

• Applying the formula:

  a_5 = 2 + (5 - 1) * 3
       = 2 + 12
       = 14
  

Step 4: Summing Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms can be calculated using the formula:

S_n = n/2 * (a_1 + a_n)

Where:

• S_n is the sum of the first n terms
• a_n is the nth term

Example

To find the sum of the first 5 terms of the sequence 2, 5, 8, 11, 14:

• First, find the 5th term (calculated as 14 above).
• Then apply the sum formula:

  S_5 = 5/2 * (2 + 14)
      = 5/2 * 16
      = 5 * 8
      = 40
  

Conclusion

In this guide, you learned about arithmetic sequences and series, including how to identify them, calculate specific terms, and sum the terms. These foundational concepts will serve you well as you continue your studies in mathematics. To deepen your understanding, consider practicing with different sequences and solving problems related to arithmetic series.