Lecture Notes: Geometric Sequences

Introduction

• Speaker: Alvin Joel Partezan
• Topic: Geometric Sequence
• Institution: Mount Laumit Sangguansang, Greater Aginado

Definition of Geometric Sequence

• A sequence where each term is obtained by multiplying the previous term by a constant number.
• Common Ratio (r): The constant number used in the sequence.
• Formula for Common Ratio: ( r = \frac{a_2}{a_1} )
• Formula for Geometric Sequence: ( a_n = a_1 \cdot r^{(n-1)} )

Example Problems

Example 1: Finding the Sixth Term

• Given Sequence: 2, 6, 13, 16, 17
• Objective: Find the sixth term.
• Steps:
  1. Find the common ratio: ( r = \frac{6}{2} = 3 )
  2. Use the ratio to generate sequence:
     • 2, 6, 18, 54, 162, 486
  3. Sixth Term: 486

Example 2: Finding the Tenth Term

• Given Sequence: Starting value ( a_1 = 2 ), ratio ( r = 3 )
• Objective: Find the 10th term.
• Steps:
  1. Formula: ( a_n = a_1 \cdot r^{(n-1)} )
  2. Substitute values: ( a_{10} = 2 \cdot 3^{(10-1)} )
  3. Simplify:
     • Calculate ( 3^9 = 19,683 )
     • Multiply: ( 2 \cdot 19,683 = 39,366 )
  4. Tenth Term: 39,366

Example 3: Finding the Fifth Term

• Given Sequence: 3, 6, 12
• Objective: Find the fifth term.
• Steps:
  1. Find the common ratio: ( r = \frac{6}{3} = 2 )
  2. Generate sequence using the common ratio:
     • 3, 6, 12, 24, 28
  3. Substitute into formula:
     • ( a_5 = 3 \cdot 2^{(5-1)} )
  4. Simplify:
     • Calculate ( 2^4 = 16 )
     • Multiply: ( 3 \cdot 16 = 48 )
  5. Fifth Term: 48

Conclusion

• The geometric sequence is defined by a common ratio that is consistent across terms.
• The formula ( a_n = a_1 \cdot r^{(n-1)} ) is crucial for finding specific terms in the sequence.

Final Note

• Remember to practice solving geometric sequences to become more familiar with the concept.
• Thanks for listening!