Lecture on Arithmetic and Geometric Sequences

Introduction

• Focus on arithmetic sequences.
• Comparison between arithmetic and geometric sequences.

Arithmetic vs. Geometric Sequences

• Arithmetic Sequence Example: 3, 7, 11, 15, 19, 23, 27
  • Constant addition: Common difference (d = 4).
• Geometric Sequence Example: 3, 6, 12, 24, 48, 96, 192
  • Constant multiplication: Common ratio (r = 2).

Arithmetic and Geometric Means

• Arithmetic Mean:
  • Formula: ((a + b) / 2)
  • Example: Between 3 and 11 is 7.
• Geometric Mean:
  • Formula: (\sqrt{a \times b})
  • Example: Between 3 and 12 is 6.

Finding Terms in Sequences

• Arithmetic Sequence Formula:
  • (a_n = a_1 + (n - 1) \cdot d)
  • Example: 5th term for sequence starting with 3 and common difference 4 is 19.
• Geometric Sequence Formula:
  • (a_n = a_1 \cdot r^{(n-1)})
  • Example: 6th term for sequence starting with 3 and common ratio 2 is 96.

Partial Sums

• Arithmetic Sequence Partial Sum:
  • (S_n = (a_1 + a_n) / 2 \times n)
  • Example: Sum of first 7 terms of sequence starting with 3 and ending with 27 is 105.
• Geometric Sequence Partial Sum:
  • (S_n = a_1 \times \frac{1 - r^n}{1 - r})
  • Example: Sum of first 6 terms of sequence starting with 3 and ratio 2 is 189.

Sequence vs. Series

• Sequence: List of numbers.
• Series: Sum of numbers in a sequence.
• Finite vs. Infinite:
  • Finite: Has a start and end.
  • Infinite: Continues indefinitely.

Practice Problems

• Identify if a series or sequence is arithmetic, geometric, or neither.
• Determine if series/sequence is finite or infinite.
• Calculate common differences or ratios to classify sequences.

Writing General Formulas

• Arithmetic Sequence:
  • Identify first term and common difference.
  • Example: Sequence 5, 14, 23 has formula (a_n = 9n - 4).
• Recursive Formulas:
  • Use previous term to find the next.

Calculating Sums and Terms

• Sum of Natural Numbers: Use partial sum formula to find sums such as the first 300 natural numbers.
• Sum of Even/Odd Numbers: Determine terms and use partial sum formula.

Conclusion

• Ability to identify and work with arithmetic and geometric sequences and series.
• Use formulas to solve real-world problems efficiently.