Arithmetic and Geometric Sequences and Series

Types of Sequences

• Arithmetic Sequence: Numbers progress by adding a constant value (common difference).

  • Example: 3, 7, 11, 15, 19, 23, 27

• Geometric Sequence: Numbers progress by multiplying a constant value (common ratio).

  • Example: 3, 6, 12, 24, 48, 96, 192

Identifying Sequences

• Arithmetic Sequence: Common Difference (d)$\rightarrow$ Add constant value

  • $a_{n} = a_{1} + (n-1)d$
  • Example calculation: 5th term $a_{5} = a_1 + 4d = 3 + 4 imes 4 = 19$

• Geometric Sequence: Common Ratio (r)$\rightarrow$ Multiply constant value

  • $a_{n} = a_{1} \times r^{(n-1)}$
  • Example calculation: 6th term $a_{6} = 3 \times 2^{5} = 96$

Arithmetic and Geometric Mean

• Arithmetic Mean: Average of two numbers $(a+b)/2.

  • Example:
    • Mean of 3 and 11: $(3 + 11) / 2 = 7$ (middle number of the sequence).
    • Mean of 7 and 23: $(7 + 23) / 2 = 15$

• Geometric Mean: Square root of product of two numbers.

  • Example:
    • Mean of 3 and 12: $\sqrt{3 \times 12} = 6$
    • Mean of 6 and 96: $\sqrt{6 \times 96} = 24$

Formulas for Sequences

• Nth Term of Arithmetic Sequence:

  • Formula: $a_{n} = a_{1} + (n-1)d$

• Nth Term of Geometric Sequence:

  • Formula: $a_{n} = a_{1} \times r^{(n-1)}$

Series and Sums

• Partial Sum of Arithmetic Sequence

  • Formula: $S_n = \frac{n}{2} (a_{1} + a_{n}) $

• Partial Sum of Geometric Sequence

  • Formula: $S_{n} = a_{1} \times \frac{1-r^n}{1-r}$

Sequences vs Series

• Sequence: List of numbers
• Series: Sum of numbers in a sequence

Types of Sequences and Series

• Finite Sequence: Has both a beginning and an end
• Infinite Sequence: Goes on indefinitely
• Finite Series: Sum of a finite sequence
• Infinite Series: Sum of an infinite sequence

Practice Problems

1. Pattern Recognition: Sequence or Series

• Identify if a pattern is a sequence or series, and if it is finite or infinite. Determine if it is arithmetic, geometric, or neither.
  • Example patterns to recognize:
    • 4, 7, 10, 13, 16, 19 $ ightarrow$ Arithmetic Sequence
    • 4, 8, 16, 32 $ ightarrow$ Geometric Sequence
    • 5 + 9 + 13 + 17 $ ightarrow$ Arithmetic Series
    • 2 + 6 + 18 $ ightarrow$ Geometric Series

2. Calculate First Terms of Sequence

• Given $a_n = 3n - 7$
  • First four terms: $a_1 = -4$, $a_2 = -1$, $a_3 = 2$, $a_4 = 5$
  • Common difference: 3

3. Find Next Terms in Arithmetic Sequence

• Given sequence: 15, 22, 29, 36
  • Common difference: 7
  • Next terms: 43, 50, 57

4. First Five Terms Given a1 and d

• Given $a_1 = 29$, $d = -4$
  • First five terms: 29, 25, 21, 17, 13

5. Recursive Formula for Sequence

• Given recursive formula and first term, find next terms:
  • Example: $a_n = a_{n-1} + 4$ with $a_1 = 3$
  • First five terms: 3, 7, 11, 15, 19

6. General or Explicit Formula for Sequences

• Example general formula: $a_n = 6n + 2$
  • Example for sequences with fractions: Separate numerator and denominator sequences
    • $a_n = (n+1) / (2n+1)$

7. Find Nth Term Formula and Sum of Sequence

• Example details for sequences:
  • Formula for nth term: $a_n = 9n - 4$
  • Sum: $S_{10} = 455$

8. Sum of First 300 Natural Numbers

• Sum of natural numbers formula: $S_{n} = \frac{n(n+1)}{2}$
  • Example: Sum of first 300 numbers $\rightarrow S_{300} = 45150$

9. Sum of Even/Odd Numbers in Sequence

• Sum of even numbers between 2 to 100 $ ightarrow 2550$
  • Sum of odd numbers between 21 and 75 $ ightarrow 1344`