How do you find geometric numbers?

Finding geometric numbers (specifically, terms in a geometric sequence) involves understanding the properties of geometric sequences and utilizing the common ratio. The short video snippet primarily addresses finding the common ratio, a key component in determining subsequent terms.

Here's a breakdown of how to find terms in a geometric sequence:

Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r).

• First term: a₁
• Second term: a₂ = a₁ * r
• Third term: a₃ = a₂ r = a₁ r²
• nth term: a_n = a₁ * r^(n-1)

Steps to Find Geometric Numbers (Terms)

1. Identify the first term (a₁): This is the starting point of the sequence.

2. Determine the common ratio (r): The video snippet emphasizes how to find 'r'. It's found by dividing any term by its preceding term. For example:

   • r = a₂ / a₁
   • r = a₃ / a₂
   • And so on...

   Example: If a₁ = 2 and a₂ = -8, then r = -8 / 2 = -4

3. Use the formula to find a specific term (a_n): Once you know a₁ and r, you can find any term in the sequence using the formula: a_n = a₁ * r^(n-1)

   Example: To find the 5th term (a₅) of the sequence where a₁ = 2 and r = -4:

   a₅ = 2 (-4)^(5-1) = 2 (-4)⁴ = 2 * 256 = 512

Example

Let's say you have the sequence: 3, 6, 12, 24,...

1. a₁ = 3
2. r = 6 / 3 = 2 (You could also check: 12/6 = 2, 24/12 = 2)
3. To find the 7th term: a₇ = 3 2^(7-1) = 3 2⁶ = 3 * 64 = 192

Summary

To find terms in a geometric sequence, you need to identify the first term and the common ratio. The common ratio is found by dividing a term by its previous term. Once you have these, you can use the formula a_n = a₁ * r^(n-1) to determine any term in the sequence.