Arithmetic and Geometric Sequences Lecture Notes

Understanding Sequences

• Arithmetic Sequence: Sequence where each term after the first is generated by adding a constant, known as the common difference, to the previous term.
  • Example: 3, 7, 11, 15... (Common difference = 4)
• Geometric Sequence: Sequence where each term after the first is generated by multiplying the previous term by a constant, known as the common ratio.
  • Example: 3, 6, 12, 24... (Common ratio = 2)

Patterns in Sequences

• Arithmetic Pattern: Based on addition or subtraction.
• Geometric Pattern: Based on multiplication or division.

Calculating Means

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

Formulas for Sequences

• Nth Term of Arithmetic Sequence: ( a_n = a_1 + (n-1)d )
• Nth Term of Geometric Sequence: ( a_n = a_1 \times r^{(n-1)} )

Calculating Terms

• Fifth Term of Arithmetic Sequence: Example with initial term 3 and common difference 4, fifth term is 19.
• Sixth Term of Geometric Sequence: Example with initial term 3 and common ratio 2, sixth term is 96.

Partial Sums of Sequences

• Arithmetic Sequence Sum:
  • Formula: ( S_n = \frac{a_1 + a_n}{2} \times n )
  • Example for first 7 terms: Sum is 105.
• Geometric Sequence Sum:
  • Formula: ( S_n = a_1 \times \frac{1 - r^n}{1 - r} )
  • Example for first 6 terms: Sum is 189.

Sequences vs. Series

• Sequence: A list of numbers.
• Series: The sum of the elements of a sequence.
• Finite vs. Infinite: Finite has clear start and end; infinite continues indefinitely.

Identifying Sequences and Series

• Check for common difference (arithmetic) or common ratio (geometric).
• Determine if finite/infinite, sequence/series.

Writing Formulas

• General Formula for Arithmetic Sequence:
  • Need first term and common difference.
  • Example: For 8, 14, 20... formula is ( a_n = 6n + 2 ).
• Example with Fractions: Break into separate sequences for numerator and denominator.

Practice Problems

• Determine sequence type and nature (finite/infinite).
• Write first few terms given a formula.
• Find common difference/ratio and use formulas to solve problems.

Additional Concepts

• Sum of Natural Numbers: Example using a series of numbers from 1 to 300.
• Sum of Even/Odd Numbers: Calculate using series and formulas.

These notes summarize the key points of arithmetic and geometric sequences, their differences, how to calculate terms, means, and sums, and how to distinguish between sequences and series. The formulas provided are crucial for solving related problems.