Lecture on Geometric Sequences

Introduction

• Review of previous lesson on sequences with additive terms.
• Current focus: Sequences where terms are multiplied by a common ratio (Geometric Sequences).

Key Concepts

1. Geometric Sequence

   • A sequence in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ((r)).

2. Common Ratio ((r))

   • Defined by dividing any term by the previous term.
   • Example calculations:
     • (8/2 = 4)
     • (9/(-3) = -3)
     • ((1/2)/1 = 1/2)

Examples and Exercises

• Example 1: Sequence (1, 2, 4, 8)

  • Common ratio: (r = 2)
  • Next term: (8 \times 2 = 16)

• Example 2: Sequence (80, 20, 5)

  • Common ratio: (r = 1/4)
  • Next term: (5 \times 1/4 = 5/4)

• Example 3: Sequence (2, -8, 32, -128)

  • Common ratio: (r = -4)
  • Next term: (-128 \times -4 = 512)

Identifying Geometric Sequences

• Example 1: (5, 20, 80, 320)

  • Common ratio: (r = 4)
  • Conclusion: Geometric sequence.

• Example 2: (7\sqrt{2}, 5\sqrt{2}, 3\sqrt{2}, \sqrt{2})

  • No common ratio.
  • Conclusion: Not a geometric sequence.

• Example 3: (5, -10, 20, -40)

  • Common ratio: (r = -2)
  • Conclusion: Geometric sequence.

• Example 4: (10/3, 10/6, 10/9, 10/15)

  • No consistent common ratio.
  • Conclusion: Not a geometric sequence.

Formula for the nth Term

• Formula: (a_n = a_1 \times r^{n-1})

  • (a_1) = first term
  • (r) = common ratio ((r \neq 0))
  • (n) = number of terms

• Example: Find the 10th term of (8, 4, 2, 1)

  • (r = 1/2), (a_1 = 8)
  • (a_{10} = 8 \times (1/2)^9 = 1/64)

Further Exercises

1. Sequence (3, 12, 48)

   • Missing term: (192, 768)

2. Sequence with missing terms: (_, _, 32, 64, 128)

   • Calculation involved division to find missing terms.

Application Problem

• Problem: Infection pattern of measles growing geometrically.
  • Given (4, 8, 16) for first three days.
  • Find infections on the 6th day.
  • Solution: (128) people infected.

Conclusion

• Reinforcement of concepts with examples and exercises.
• Encouragement to like, subscribe, and follow for more tutorials.

Note: Geometric sequences are characterized by a consistent multiplicative progression using a common ratio, making calculations predictable and formula-based.