This section covers geometric sequences, their properties, and methods for calculating terms and means within these sequences.

Definition: A geometric sequence is characterized by a constant ratio between consecutive terms, known as the common ratio 'r'.

Vocabulary: Common ratio $r$ is the constant value by which each term is multiplied to obtain the next term.

Example: In the sequence 1, 3, 9, 27, the common ratio is 3.

Highlight: The explicit formula for geometric sequences is an = a₁$r$^$n-1$, where a₁ is the first term and r is the common ratio.

Example: Finding 3 geometric means between 2 and 162:

• Initial terms: 2 and 162
• Calculated means create a sequence with constant ratio
• Process involves finding intermediate terms that maintain the geometric progression

This section explores the fundamental concepts of arithmetic sequences, where consecutive terms maintain a constant difference. The content details how to identify and work with these sequences effectively.

Definition: An arithmetic sequence is a sequence where the difference between successive terms remains constant, denoted as the common difference 'd'.

Vocabulary: Common difference $d$ represents the constant value added to each term to get the next term in an arithmetic sequence.

Example: In a sequence where terms progress as: 2, 4, 6, 8..., the common difference is +2.

Highlight: The explicit formula for finding any term in an arithmetic sequence is an = a₁ + d$n-1$, where a₁ is the first term and n is the term number.

Example: Finding arithmetic means between -30 and -45:

• Starting with -30
• Common difference calculated as -5
• Sequence progresses: -30, -35, -40, -45