Geometric Sequences

Introduction

• Focus on sequences where a number is multiplied to get the next term.
• Definition of geometric sequence and identification of the common ratio.
• Strategies to find the nth term and solve related problems.

Key Concepts

Common Ratio (r)

• The common ratio in a geometric sequence is determined by dividing any term by the preceding term.
• Notation for common ratio is represented by the letter r.

Identifying Common Ratio and Next Terms

1. Example 1: Sequence: 1, 2, 4, 8

   • Common Ratio: 2 (8 / 4)
   • Next Term: 16 (8 * 2)

2. Example 2: Sequence: 80, 20, 5

   • Common Ratio: 1/4 (20 / 80)
   • Next Term: 5/4 (5 * 1/4)

3. Example 3: Sequence: 2, -8, 32, -128

   • Common Ratio: -4 (-8 / 2)
   • Next Term: 512 (-128 * -4)*

Determining Geometric Sequences

• Check if a sequence is geometric by ensuring all ratios between consecutive terms are equal.
• Example Analysis:
  1. Sequence: 5, 20, 80
     • Common Ratio: 4 (20 / 5, 80 / 20)
     • Conclusion: Geometric Sequence
  2. Sequence: 5√2, 3√2, ...
     • Common Ratio: 5/7
     • Conclusion: Not a Geometric Sequence
  3. Sequence: 5, -10, 20, -40
     • Common Ratio: -2
     • Conclusion: Geometric Sequence
  4. Sequence: 10/3, 10/6, 10/9, 10/15
     • Different common ratios observed
     • Conclusion: Not a Geometric Sequence

nth Term of a Geometric Sequence

• Formula to find the nth term:
  [ a_n = a_1 \cdot r^{n-1} ]
• Where:
  • ( a_1 ) = first term
  • ( r ) = common ratio
  • ( n ) = number of terms

Example Calculation

• Find the 10th term of the sequence 8, 4, 2, 1:
  • First term (a1) = 8
  • Common Ratio (r) = 1/2
  • 10th term calculation: [ 8 \cdot (1/2)^{10-1} ]
  • Result: 1/64

Activities and Exercises

1. Find the missing term in the sequence: 3, 12, 48
   • Result: 192 (next term)
2. Find the missing term in the sequence: __, 32, 64, 128
   • Result: 16
3. Application Problem: Infection growth in an outbreak:
   • Sequence: 4, 8, 16 (first three days)
   • Find infected on sixth day using the formula:
     [ 4 \cdot 2^{(6-1)} ]
   • Result: 128 infected individuals on the sixth day.__

Conclusion

• Review the process of finding common ratios and nth terms.
• Engage with additional exercises for practice.