Lesson 2: Geometric Sequences and Series

Geometric Sequence

• A sequence where each term is found by multiplying the previous term by a constant (common ratio).
• Example: Sequence 2, 6, 18
  • Common ratio ( r = \frac{6}{2} = 3 )

Common Ratio

• Obtained by dividing the second term by the first term.
• Example calculation:
  • ( a_2 / a_1 = 6 / 2 = 3 )

Finding Terms

• To find the nth term, use the formula:
  • ( a_n = a_1 \times r^{(n-1)} )
  • Example: To find 5th term given sequence starts with 2 and ( r = 3 )
    • ( a_5 = 2 \times 3^{4} = 2 \times 81 = 162 )

Geometric Mean

• The term between any two terms in a geometric sequence.
• Formula: ( GM = \sqrt{a \times b} )
  • Example: Find GM between 2 and 8
    • ( GM = \sqrt{2 \times 8} = \sqrt{16} = 4 )

Geometric Series

• The sum of a geometric sequence.
• Formula for sum of first n terms:
  • ( S_n = a_1 \times \frac{r^n - 1}{r - 1} )
  • Example: Sum of first 8 terms of sequence 1, 3, 9, 27
    • ( S_8 = 1 \times \frac{3^8 - 1}{3 - 1} )
    • Calculate ( 3^8 = 6561 )
    • ( S_8 = \frac{6561 - 1}{2} = 3280 )

Infinite Geometric Series

• Sum of an infinite geometric sequence.
• Formula:
  • ( S_{\infty} = \frac{a_1}{1 - r} )
  • Condition: (|r| < 1)
  • Example: Sequence 8, -2, 1/2, -1/8
    • ( S_{\infty} = \frac{8}{1 - (-1/4)} = \frac{8}{5/4} = \frac{32}{5} )

Practical Example: Pendulum

• First swing length 10 decimeters; each successive swing is 2/3 of the preceding.
• Total distance before rest calculated using sum formula for infinite geometric series.

Recap

• The geometric sequence finds terms through multiplication by a common ratio.
• Geometric mean finds a middle term using square roots.
• Geometric series sums terms of a sequence.
• Infinite geometric series deals with sequences that continue indefinitely under specific conditions.