Geometric Sequences and Their Application

Definition: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Daily Objective: Students will be able to use the geometric sequences formula to find the nth term.

Arithmetic Sequence:
- Defined as a sequence where the difference between consecutive terms is constant.
- Example: 5, 10, 15 (common difference = +5)
- Example: 12, 6, 0, -6, -12 (common difference = -6)
Geometric Sequence:
- Characterized by the ratio of consecutive terms being constant. The ratio is known as the common ratio.
- Example: 2, 4, 8, 16, 32 (here the common ratio is 2, each term is multiplied by 2)
- This does not qualify as an arithmetic sequence because it does not have a constant difference.

Geometric Sequence Formula: an = a1 imes r^{(n-1)}
- Where:
- a_n is the nth term of the sequence,
- a_1 is the first term,
- r is the common ratio,
- n is the term number.

Find the 13th term in the sequence: 4, 12, 36, …
- Identify the first term and common ratio:
- a_1 = 4
- Sequence pattern: 4, 12, 36 (after examining, multiply by 3 to get from 4 to 12, and from 12 to 36)
- Common Ratio r = 3
- Using the formula:
  a_n = 4 imes 3^{(n-1)}
- To find the 13th term:
  a_{13} = 4 imes 3^{(13-1)} = 4 imes 3^{12}

A clear understanding of geometric and arithmetic sequences is essential, especially in recognizing their defining characteristics (common difference vs. common ratio).
Apply the appropriate formulas to find terms in these sequences, facilitating problem-solving in related mathematical contexts.