Overview
This lecture explains geometric means, including their definition, formulas, and step-by-step examples on finding missing terms in geometric sequences.
Geometric Means Overview
- Geometric means are the terms between the first and last term in a geometric sequence.
- A geometric sequence is a list of numbers with a constant ratio between consecutive terms.
- The general formula for the nth term is ( a_n = a_1 \times r^{n-1} ).
Example 1: One Geometric Mean (5, , 20)
- To find a single geometric mean between two numbers, multiply them and take the square root: ( \sqrt{5 \times 20} = 10 ).
- Alternatively, use the formula: set up ( 20 = 5 \times r^2 ), solve for ( r = 2 ), then ( 5 \times 2 = 10 ).
- The geometric mean between 5 and 20 is 10.
Example 2: Two Geometric Means (2, __, __, 686)
- Use ( a_4 = a_1 \times r^{3} ): ( 686 = 2 \times r^3 ).
- Divide both sides by 2: ( 343 = r^3 ), so ( r = 7 ).
- Second term: ( 2 \times 7 = 14 ).
- Third term: ( 14 \times 7 = 98 ).
- The sequence is 2, 14, 98, 686.
Example 3: Fractions (3, __, __, 1/9)
- Use ( a_4 = a_1 \times r^3 ): ( 1/9 = 3 \times r^3 ).
- Divide both sides by 3: ( 1/27 = r^3 ), so ( r = 1/3 ).
- Second term: ( 3 \times 1/3 = 1 ).
- Third term: ( 1 \times 1/3 = 1/3 ).
- The sequence is 3, 1, 1/3, 1/9.
Key Terms & Definitions
- Geometric Mean — The term(s) inserted between the first and last terms of a geometric sequence.
- Geometric Sequence — A sequence where each term is multiplied by a constant ratio.
- Common Ratio (r) — The fixed number multiplied to each term to get the next term.
- nth Term Formula — ( a_n = a_1 \times r^{n-1} ).
Action Items / Next Steps
- Practice similar problems on finding geometric means in geometric sequences.
- Review geometric sequence formulas and their applications.