مراجعة المتتاليات + تمرين رائع (حل باك 2024) والاهم شرح اروع🔥

Introduction

This tutorial is designed to provide a comprehensive understanding of sequences, a crucial topic in mathematics, especially for those preparing for exams like the Baccalaureate in 2024. In this guide, we will explore the fundamentals of sequences and work through an engaging exercise that will solidify your understanding.

Step 1: Understanding Sequences

• Definition of a Sequence: A sequence is an ordered list of numbers that follow a specific pattern.

Types of Sequences

• Arithmetic Sequences: Each term is obtained by adding a constant to the previous term.
• Geometric Sequences: Each term is obtained by multiplying the previous term by a constant.
• Notation: Sequences are usually denoted as ( a_n ), where ( n ) represents the position of the term in the sequence.

Practical Advice

Familiarize yourself with common formulas for sequences

• For an arithmetic sequence: ( a_n = a_1 + (n-1) \cdot d ), where ( d ) is the common difference.
• For a geometric sequence: ( a_n = a_1 \cdot r^{(n-1)} ), where ( r ) is the common ratio.

Step 2: Analyzing Sequence Problems

• Identify the Sequence Type: Determine whether it’s arithmetic or geometric based on the pattern.

Finding the nth Term

• Use the appropriate formula for calculating the nth term based on the identified sequence type.

Practical Advice

• Double-check calculations by plugging in known values to ensure your formula is correct.

Step 3: Solving a Sequence Problem

• Example Problem: Solve for the 10th term of the sequence defined by ( a_1 = 3 ) and ( d = 5 ).

  Solution

  1. Identify the sequence type (arithmetic).
  2. Use the formula:

     a_n = a_1 + (n-1) * d
     

  3. Substitute the values:

     a_{10} = 3 + (10-1) * 5
            = 3 + 45
            = 48
     

  • The 10th term is 48.

Common Pitfalls to Avoid

• Misidentifying the type of sequence can lead to incorrect calculations. Always verify the pattern before applying formulas.

Step 4: Practice Exercise

• Exercise: Given the sequence defined by ( a_1 = 7 ) and ( r = 2 ), find the 8th term.

  Steps to Solve

  1. Identify the sequence type (geometric).
  2. Apply the geometric sequence formula:

     a_n = a_1 * r^{(n-1)}
     

  3. Substitute the values:

     a_8 = 7 * 2^{(8-1)}
         = 7 * 128
         = 896
     

  • The 8th term is 896.

Conclusion

In this tutorial, we covered the basics of sequences, learned how to identify types, calculated specific terms, and practiced with real problems. Mastery of sequences is essential for your upcoming exams, particularly the Baccalaureate in 2024. Continue practicing with different sequence problems to enhance your understanding and confidence.