Mathematics - Sequences and Series Study Notes

Geometric Sequences Formula

• The formula for a geometric sequence is expressed as:

  • $a<em>n = a</em>1 imes r^{n-1}$

  • Where:

    • $a_n$ = the nth term of the sequence

    • $a_1$ = the initial term

    • $r$ = the common ratio (what each term is multiplied by)

    • $n$ = the term number

  • An alternative expression, if the zero term is given, is:

  • $a<em>n = a</em>0 imes r^n$

Differences Between Sequence and Series

• Sequence:

  • A sequence is defined as a list of numbers arranged in a specific order.

• Series:

  • A series involves the summation of the elements of a sequence.

  • In this course, the emphasis is placed on series rather than on sequences.

  • This focus aligns with learning about summation.

Types of Sequence Formulas

• The course primarily focuses on explicit formulas, specifically:

  • Arithmetic Sequences

  • Geometric Sequences

• Recursive Definitions:

  • These definitions depend on the preceding term in the sequence.

  • They are not the main focus of this class.

• Explicit Definitions:

  • Able to input any value for n directly into the formula.

Practice Problems for Sequence Types

• In upcoming problems, students will categorize sequences as:

  • Arithmetic - sequences that have a constant difference between consecutive terms.

  • Geometric - sequences that have a constant ratio between consecutive terms.

  • Neither - sequences that do not fit into the aforementioned categories.

Solving Sequence Problems

• To solve sequence problems, students will plug values into the established formulas to identify properties of the sequence.