Arithmetic Sequences and Geometric Sequences

Introduction to Sequences

• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant (common difference).

  • Example: 3, 7, 11, 15, 19, 23, 27 (common difference of 4)

• Geometric Sequence: A sequence where the ratio between consecutive terms is constant (common ratio).

  • Example: 3, 6, 12, 24, 48, 96, 192 (common ratio of 2)

Key Differences

• Arithmetic Sequence: Pattern based on addition/subtraction.
• Geometric Sequence: Pattern based on multiplication/division.

Arithmetic Mean

• Formula: Arithmetic Mean of two numbers a and b is calculated as:

  [ \text{Mean} = \frac{a + b}{2} ]

  • Example: Mean of 3 and 11:
    [ \text{Mean} = \frac{3 + 11}{2} = 7 ]

• Example: Mean of 7 and 23:
  [ \text{Mean} = \frac{7 + 23}{2} = 15 ]

Geometric Mean

• Formula: Geometric Mean of two numbers a and b is calculated as:

  [ \text{Mean} = \sqrt{a \times b} ]

  • Example: Geometric Mean of 3 and 12:
    [ \text{Mean} = \sqrt{3 \times 12} = 6 ]

• Example: Geometric Mean of 6 and 96:
  [ \text{Mean} = \sqrt{6 \times 96} = 24 ]

Formulas for nth Term

• Arithmetic Sequence nth Term Formula:

  [ a_n = a_1 + (n - 1) \cdot d ]

  • Example: Find the 5th term in 3, 7, 11, ...
    • a_1 = 3, d = 4
    • [ a_5 = 3 + (5-1) \cdot 4 = 19 ]

• Geometric Sequence nth Term Formula:

  [ a_n = a_1 \cdot r^{(n - 1)} ]

  • Example: Find the 6th term in 3, 6, 12, ...
    • a_1 = 3, r = 2
    • [ a_6 = 3 \cdot 2^{(6-1)} = 96 ]

Partial Sums

• Arithmetic Partial Sum Formula:

  [ S_n = \frac{(a_1 + a_n)}{2} \cdot n ]

  • Example: Sum of first 7 terms in an arithmetic sequence
    • S_7 = [ \frac{(3 + 27)}{2} \cdot 7 = 105 ]

• Geometric Partial Sum Formula:

  [ S_n = \frac{a_1 \cdot (1 - r^n)}{1 - r} ]

  • Example: Sum of first 6 terms in a geometric sequence
    • S_6 = 189

Definition of Sequence vs Series

• Sequence: A list of numbers
• Series: The sum of the numbers in a sequence

Finite vs Infinite Sequences and Series

• Finite: Has a beginning and an end.
• Infinite: Continues indefinitely (e.g., 3, 7, 11, ...)

Practice Problems

1. Identify if the following is a sequence or series, finite or infinite, and type (arithmetic or geometric).

   • Example 1: 4, 7, 10, 13, 16, 19 - Sequence (Arithmetic)
   • Example 2: 4, 8, 16 - Sequence (Geometric)

2. Write the first four terms of the sequence defined by:

   [ a_n = 3n - 7 ]

   • n=1 gives -4, n=2 gives -1, n=3 gives 2, n=4 gives 5.

3. Find the sum of the first 300 natural numbers:

   • Use [ S_{300} = \frac{(1 + 300)}{2} \cdot 300 = 45,150 ]

4. Sum of all even numbers from 2 to 100:

   • S = [ \frac{2 + 100}{2} \cdot 50 = 2,550 ]

5. Sum of all odd integers from 21 to 75:

   • S = [ \frac{21 + 75}{2} \cdot 28 = 1,344 ]_

Conclusion

• Understanding and distinguishing between arithmetic and geometric sequences is key to solving related problems.
• Practice using the formulas for means, nth terms, and sums for both types of sequences.