Sequences and Arithmetic Sequences

Sequences

• A sequence is an ordered list of numbers, called terms, that follow a specific pattern or rule.
• Sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely).
• The terms of a sequence are usually denoted by a subscript, for example, $a_1, a_2, a_3, ...$ representing the first, second, third, and so on, terms.
• Sequences can be defined explicitly, meaning there's a formula to calculate any term directly from its position. For example, $a_n = 2n + 1$, which gives the nth term as an expression of $n$.
• Alternatively, a sequence can be defined recursively, which involves stating the first term and relating subsequent terms to previous ones. For example, $a_1 = 1, a_{n+1} = a_n + 2$, for $n \ge 1$.

Arithmetic Sequences

• An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
• This constant difference is called the common difference, often denoted by 'd'.
• The general form of an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term.
• Examples include 2, 5, 8, 11, ... (common difference is 3)
• Recognizing this pattern helps to quickly identify an arithmetic sequence.

Geometric Sequences

• A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed non-zero number.
• This fixed number is called the common ratio, often denoted by 'r'.
• The general form is $a_n = a_1 \times r^{n-1}$, where $a_1$ is the first term.
• Examples include 2, 6, 18, 54, ... (common ratio is 3)
• The common ratio is calculated by dividing any term by the previous term.

Series

• A series is the sum of the terms in a sequence.
• Finite series are sums of a fixed number of terms.
• Infinite series are sums of infinitely many terms.
• The sum of the first n terms of a series is often denoted by $S_n$.

Arithmetic Series

• An arithmetic series is the sum of an arithmetic sequence.
• The sum of the first 'n' terms of an arithmetic series can be calculated using the formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $a_1$ is the first term, and $a_n$ is the nth term. Alternatively, $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.

Geometric Series

• A geometric series is the sum of a geometric sequence.
• The sum of the first 'n' terms of a geometric series is given by the formula $S_n = \frac{a_1 (1 - r^n)}{1 - r}$, where $a_1$ is the first term, and 'r' is the common ratio (assuming r ≠ 1).

Finite Series

• For finite series, the sum can be explicitly calculated by adding the terms.

Infinite Series

• An infinite series can converge (have a finite sum) or diverge (have an infinite or undefined sum).
• Convergence is determined by the value of the common ratio 'r' in geometric sequences.

Convergence of Geometric Series

• An infinite geometric series converges if and only if $|r| < 1$.
• The sum of an infinite convergent geometric series is given by $S = \frac{a_1}{1 - r}$.

Sigma Notation

• Sigma notation is a concise way to represent a series. It uses the Greek capital letter 'Σ' to represent summation.
• For example, $\sum_{i=1}^{n} a_i$ represents the sum of the terms $a_1, a_2, ..., a_n$.

Applications

• Sequences and series have wide applications in various fields, including finance (calculating compound interest), physics (modeling oscillations), and computer science (generating patterns).