Arithmetic and Geometric Sequences

Distinction Between Sequences

• Arithmetic Sequence: A sequence where each term is obtained by adding a fixed common difference to the previous term.
  • Example: 3, 7, 11, 15, 19, 23, 27 (common difference = 4)
• Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a fixed common ratio.
  • Example: 3, 6, 12, 24, 48, 96, 192 (common ratio = 2)

Calculating Means

• Arithmetic Mean: The average of two numbers (a + b) / 2.
  • Example: The mean of 3 and 11 is 7.
• Geometric Mean: The square root of the product of two numbers ( \sqrt{a \times b} ).
  • Example: The mean of 3 and 12 is 6.

Formulas

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

Calculating Specific Terms

• Arithmetic Example: 5th term with ( a_1 = 3 ) and ( d = 4 ): ( a_5 = 3 + 4 \times 4 = 19 )
• Geometric Example: 6th term with ( a_1 = 3 ) and ( r = 2 ): ( a_6 = 3 \times 2^5 = 96 )

Partial Sums

• Arithmetic Series: ( S_n = \frac{a_1 + a_n}{2} \times n )
• Geometric Series: ( S_n = a_1 \times \frac{1 - r^n}{1 - r} )

Examples

• Arithmetic Sum: Sum of first 7 terms: ( S_7 = \frac{3 + 27}{2} \times 7 = 105 )
• Geometric Sum: Sum of first 6 terms: ( S_6 = 3 \times \frac{1 - 2^6}{1 - 2} = 189 )

Sequence vs. Series

• Sequence: List of numbers, finite or infinite.
• Series: Sum of numbers in a sequence, finite or infinite.
  • Example: 4, 7, 10, 13, 16, 19 (sequence) vs. 4 + 7 + 10 + 13 + 16 + 19 (series)

Practice Problems

• Determine if a list is a sequence or series, finite or infinite, and arithmetic or geometric.
  • Example: 4, 7, 10, 13, 16 (finite arithmetic sequence)

Writing Terms and Formulas

• Write First Few Terms: Use given formulas or sequences to calculate terms.
  • Example: ( a_n = 3n - 7 ): First terms are -4, -1, 2, 5
• Write General Formula: Use first term and common difference/ratio.
  • Example: For ( a_n = 6n + 2 )

Recursive Formulas

• Use previous terms to calculate next terms.
  • Example: ( a_n = 3a_{n-1} + 2 )_

Additional Examples

• Sum of Natural Numbers: Calculate using arithmetic series formula.
  • Example: Sum of first 300 natural numbers is 45,150.
• Odd/Even Sum Calculation: Use formulas to find sum of specific sequences.
  • Example: Sum of even numbers from 2 to 100 is 2,550.

These notes provide an overview and practical examples for calculating terms and sums in arithmetic and geometric sequences and series, helping in understanding how to work with both finite and infinite sequences.