Formulating Geometric Sequence Rule 5.2

Mathematics Transcription Notes

Sequence Patterns and Observations

• Motivation: Observe the following patterns:

  • Sequence 1: 1, 2, 4, 8, 16, ___

  • Sequence 2: 500, 200, 80, 32, 12.8, ___

  • Sequence 3: 1/2, 1/6, 1/18, 1/54, ___

• Guide Questions:

  • What operation is repeated?

  • What do you notice about the ratio between terms?

Definition of Geometric Sequence

• Geometric sequence: A sequence in which the ratio between consecutive terms is constant.

  • This constant ratio is referred to as the COMMON RATIO (r).

  • Each term is formed by multiplying the previous term by the common ratio (r).

Comparison: Arithmetic vs Geometric Sequence

• Arithmetic Sequence:

  • Definition: Involves adding a constant number to each term.

• Geometric Sequence:

  • Definition: Involves multiplying each term by a constant number (r).

• Key Idea: The difference in nature between:

  • Constant difference in arithmetic sequences.

  • Constant ratio in geometric sequences.

Types of Geometric Sequences

• Finite Geometric Sequences:

  • Example: 1, 5, 25, 125, 625

  • Characteristics: Contain a definite number of terms.

• Infinite Geometric Sequences:

  • Example: 80, 40, 20, 10, 5, …

  • Characteristics: Continue indefinitely, possessing no endpoint.

Examples of Determining Geometric Sequences

• Example 1: Is it Geometric?

  • Given: 128, 64, 24, 12, 6

  • Computation of ratios:

  • Ratio 1: 64 ÷ 128 = 1/2

  • Ratio 2: 24 ÷ 64 = 3/8

  • Conclusion: Ratios are NOT equal, thus Not a geometric sequence.

• Example 2: Is it Geometric?

  • Given: -2, 6, -18, 54

  • Computation of common ratio:

  • Ratio 1: 6 ÷ (-2) = -3

  • Ratio 2: -18 ÷ 6 = -3

  • Conclusion: Common ratio r = -3, thus Geometric sequence.

Formulas for Geometric Sequences

• 1. Recursive Form Formula:

  • General form: aₙ = r · aₙ₋₁ (for n ≥ 2)

  • Example 1: 1, 2, 4, 8, 16

  • First term, a₁ = 1

  • Common ratio, r = 2

  • Each term = Previous term × 2

• 2. Explicit Form Formula:

  • General form: aₙ = a₁ · rⁿ⁻¹ (for n ≥ 1)

  • Used to find any term directly.

  • Example 2: 4, 12, 36, 108, 324

  • First term, a₁ = 4

  • Common ratio, r = 3

Example Using Explicit Formula

• Example 1: Find the 6th term of the sequence 4, 12, 36, 108, 324.

  • Given values: a₁ = 4, r = 3, n = 6

  • Calculation of the sixth term:

  • a₆ = 4(3)⁵

  • a₆ = 4(243)

  • a₆ = 972

  • Formula Reference: aₙ = a₁ · rⁿ⁻¹ (for n ≥ 1)

Lesson Summary

• A geometric sequence is characterized by constant ratio (r).

• Formulas:

  • Recursive: aₙ = r · aₙ₋₁

  • Explicit: aₙ = a₁ · rⁿ⁻¹

• Always identify the first term (a₁) and common ratio (r) for proper understanding and manipulation of the sequences.