Lecture on Arithmetic and Geometric Sequences

Key Concepts

Arithmetic and Geometric Sequences

• Arithmetic Sequence:

  • Defined by a common difference.
  • Example: 3, 7, 11, 15, 19, 23, 27.
  • Common Difference: Add a constant value to get the next term.
  • Formula for nth term: $a_n = a_1 + (n-1) \cdot d$

• Geometric Sequence:

  • Defined by a common ratio.
  • Example: 3, 6, 12, 24, 48, 96, 192.
  • Common Ratio: Multiply by a constant value to get the next term.
  • Formula for nth term: $a_n = a_1 \cdot r^{(n-1)}$

Mean Calculations

• Arithmetic Mean:

  • Formula: $(a + b) / 2$
  • Example: Mean of 3 and 11 is $(3 + 11) / 2 = 7$

• Geometric Mean:

  • Formula: $\sqrt{a \cdot b}$
  • Example: Mean of 3 and 12 is $\sqrt{3 \cdot 12} = 6$

Calculating Terms in Sequences

Arithmetic Sequence Example

• Find the 5th term: $a_5 = a_1 + 4d$
  • Example Calculation: if $a_1 = 3$, $d = 4$: $a_5 = 3 + 4 \cdot 4 = 19$

Geometric Sequence Example

• Find the 6th term: $a_6 = a_1 \cdot r^5$
  • Example Calculation: if $a_1 = 3$, $r = 2$: $a_6 = 3 \cdot 32 = 96$

Summation of Sequences

Arithmetic Series

• Partial Sum Formula: $S_n = \frac{(a_1 + a_n)}{2} \times n$
  • Example: Sum of first 7 terms $S_7 = (3 + 27) / 2 \times 7 = 105$

Geometric Series

• Partial Sum Formula: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
  • Example: Sum of first 6 terms $S_6 = 3 \cdot \frac{1 - 64}{1 - 2} = 189$

Identifying Sequences and Series

• Sequence vs Series:
  • Sequence: List of numbers.
  • Series: Sum of numbers in a sequence.
• Finite vs Infinite:
  • Finite has definite end.
  • Infinite continues indefinitely (indicated by ...).

Practice Problem Types

• Determining sequence or series, finite or infinite, arithmetic or geometric.
• Writing sequences using recursive formulas.
• Calculating sums of specific sequence types.

Additional Examples

General Formula

• For Arithmetic Sequence: Identify first term and common difference.
• Example Formula: $a_n = 6n + 2$ derived from: first term = 8, difference = 6.

Recursive Formula Example

• Generate terms by using previous terms in the formula.
• Example for Arithmetic: $a_2 = a_1 + 4$

Application Examples

• Sum of 300 Natural Numbers:
  • Use summation formula to find sum: $S_{300}$.
• Sum of all Even/Odd Numbers within Range:
  • Identify common difference and terms, use arithmetic sum formula.

Key Takeaways

• Understand the difference between arithmetic and geometric sequences.
• Familiarize with formulas for nth terms and summation of sequence.
• Practice identifying sequence type and calculating terms and sums.