12.1 Sequences

Sequences and series are fundamental concepts in algebra, building on patterns and relationships between numbers. They provide a framework for understanding how values change over time or position, from simple arithmetic progressions to more complex mathematical structures.

These concepts are crucial for modeling real-world phenomena and solving problems in various fields. By mastering sequences and series, you'll gain powerful tools for analyzing patterns, making predictions, and calculating sums of large datasets efficiently.

Sequences and Series

Generation of initial sequence terms

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• Identify the rule or pattern that defines the sequence
  • Arithmetic sequences have a constant difference between consecutive terms (common difference)
  • Geometric sequences have a constant ratio between consecutive terms (common ratio)
  • Other sequences follow a specific formula or pattern unique to the sequence
• Apply the rule or pattern to generate the initial terms
  • For arithmetic sequences, add the common difference to the previous term to find the next term (e.g., 2, 5, 8, 11, ... with a common difference of 3)
  • For geometric sequences, multiply the previous term by the common ratio to find the next term (e.g., 3, 6, 12, 24, ... with a common ratio of 2)
  • For other sequences, follow the given formula or pattern to generate the initial terms (e.g., the Fibonacci sequence: 0, 1, 1, 2, 3, 5, ... where each term is the sum of the two preceding terms)

Derivation of nth term formula

• Analyze the pattern or relationship between the term number and the corresponding term value
• For arithmetic sequences:
  • $a_n = a_1 + (n - 1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference
  • Example: For the arithmetic sequence 3, 7, 11, 15, ..., the nth term formula is $a_n = 3 + (n - 1) \cdot 4$
• For geometric sequences:
  • $a_n = a_1 \cdot r^{n-1}$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $r$ is the common ratio
  • Example: For the geometric sequence 2, 6, 18, 54, ..., the nth term formula is $a_n = 2 \cdot 3^{n-1}$
• For other sequences, derive a formula based on the observed pattern
  • Example: For the sequence 1, 4, 9, 16, ..., the nth term formula is $a_n = n^2$
• Some sequences can be represented by a closed-form expression, which directly gives the nth term without recursion

Application of factorial notation

• Understand factorial notation: $n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$
  • Example: $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$
  • $0! = 1$ by definition
• Use factorial notation in sequence formulas or calculations when necessary
  • Example: The nth term of a sequence is given by $a_n = \frac{n!}{2^n}$
    • For $n = 3$, $a_3 = \frac{3!}{2^3} = \frac{6}{8} = \frac{3}{4}$

Calculation of partial sums

• Partial sum is the sum of a specific number of terms in a sequence
• For arithmetic sequences:
  • $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $a_n$ is the nth term, and $d$ is the common difference
  • Example: For the arithmetic sequence 2, 5, 8, 11, ..., find the sum of the first 10 terms
    • $S_{10} = \frac{10}{2}(2 + 29) = 5 \cdot 31 = 155$
• For geometric sequences:
  • $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $r$ is the common ratio
  • $S_n = a_1 \cdot n$ for $r = 1$
  • Example: For the geometric sequence 3, 6, 12, 24, ..., find the sum of the first 5 terms
    • $S_5 = \frac{3(1-2^5)}{1-2} = \frac{3(1-32)}{-1} = \frac{-93}{-1} = 93$

Expression of series in summation notation

• Summation notation is a compact way to represent the sum of terms in a series
  • $\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + ... + a_n$, where $a_i$ is the ith term and $n$ is the number of terms
• Express arithmetic series using summation notation
  • $\sum_{i=1}^{n} (a_1 + (i-1)d)$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms
  • Example: Express the arithmetic series 2 + 5 + 8 + 11 + ... + 29 using summation notation
    • $\sum_{i=1}^{10} (2 + (i-1) \cdot 3)$
• Express geometric series using summation notation
  • $\sum_{i=1}^{n} a_1 \cdot r^{i-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms
  • Example: Express the geometric series 3 + 6 + 12 + 24 + 48 using summation notation
    • $\sum_{i=1}^{5} 3 \cdot 2^{i-1}$

Behavior of Infinite Sequences and Series

• Convergence occurs when the terms of a sequence or the partial sums of a series approach a finite limit
• Divergence happens when a sequence or series does not converge to a finite limit
• The limit of a sequence is the value that the terms approach as n approaches infinity
• An infinite series is the sum of all terms in an infinite sequence
  • Some infinite series converge to a finite sum, while others diverge