How do you find the geometric sequence between two numbers?

To find a geometric sequence between two numbers, you need to determine the common ratio and then use it to calculate the terms between those numbers. Here's a breakdown of the process:

Understanding Geometric Sequences

A geometric sequence is a series 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 (r).

Steps to Find the Geometric Sequence

1. Identify the First and Last Terms: Let's say the first number is a and the last number is b, and you want to insert n terms between them to form a geometric sequence.

2. Determine the Total Number of Terms: If you insert n terms, the total number of terms will be n + 2. This includes a as the first term and b as the last term of the sequence.

3. Use the Geometric Sequence Formula: The nth term of a geometric sequence can be found using the formula:

   • a_n = a r^(n-1)*
     Where:
     • a_n is the nth term.
     • a is the first term.
     • r is the common ratio.
     • n is the term number.

4. Solve for the Common Ratio (r):

   • You know the first term a and the last term b, and the last term b is the n+2 term. So we can rewrite formula as :
   • b = a r^(n+2-1) or, b = a r⁽ⁿ⁺¹⁾
   • Divide both sides by a: b/a = r⁽ⁿ⁺¹⁾
   • Take the (n+1)th root of both sides to solve for r: r = ⁽ⁿ⁺¹⁾√(b/a)

5. Calculate the Intermediate Terms: Once you have the common ratio r, you can find all the intermediate terms by multiplying the preceding term by r.

   • The second term is a r
   • The third term is (a r) r which is a r²*
   • The fourth term is (a r²) r which is a r³*
   • And so on until you reach the last term which is b.

Example

Let’s say you want to find three geometric means between 3 and 48.

1. First Term (a): 3

2. Last Term (b): 48

3. Number of Terms to Insert (n): 3

4. Total Number of Terms: 3+2 = 5

5. Calculate the Common Ratio (r):

   • 48 = 3 r⁽³⁺¹⁾*
   • 48 = 3 r⁴*
   • 16 = r⁴
   • r = ⁴√16
   • r = 2

6. Calculate Intermediate Terms:

   • Second term: 3 * 2 = 6
   • Third term: 6 * 2 = 12
   • Fourth term: 12 * 2 = 24
   • Fifth term: 24 * 2= 48

Therefore, the geometric sequence is 3, 6, 12, 24, and 48.

According to the Reference

As highlighted in the reference video segment (17:28 to 31:19), once you have determined the first term and the common ratio, you can easily derive subsequent terms. For instance, the video shows that if the first term is -3 and the common ratio is -4, then the second term is obtained by multiplying -3 by -4, resulting in 12.