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).

Explore the concepts of sequences and arithmetic sequences, focusing on their definitions, characteristics, and rules. This quiz covers both explicit and recursive definitions, providing a solid foundation in understanding sequences in mathematics.