Geometric Sequences

Key Concepts

• Identify and describe an arithmetic sequence.
• Identify and describe a geometric sequence.
• Write the recursive formula for a sequence.
• Use the explicit formula.
• Connect geometric sequences and exponential functions.
• Apply the recursive and explicit formulas.
• Explain the formula for the sum of a finite geometric series.
• Use a finite geometric series.

Geometric Sequences

1. Arithmetic sequence 

A man is going upstairs. Height of each step is increasing constantly than the previous step. 

There can be gradual change in numbers also. 

• A number sequence in which the common difference between two consecutive terms is constant is called an arithmetic sequence. 
• Example: The common difference of the given sequence is +5. 

2. Geometric sequence 

• A sequence in which the constant ratio between two consecutive terms is constant is called a geometric sequence. 

• The common difference between the consecutive terms of the geometric sequence is not constant. 
• Example: The common ratio of terms of the sequence is 2 

3. Recursive formula for a sequence 

• We can use the recursive formula to find the next term of a geometric sequence. 

Example: Write the recursive formula for a geometric sequence 2, 10, 50, 250, … 

The constant ratio of the given sequence is 5. 

The recursive formula for a geometric sequence is a_n = r(a_n−1)

So, the recursive formula for the sequence 2, 10, 50, 250, … is a_n = 5(a_n−1)

4. Explicit formula for a sequence 

• We can use the explicit formula to find the 8th term of a geometric sequence. 

Example: What is the 10^th term of the geometric sequence 10.5, 21, 42, 84…?  

Sol: Using the explicit formula, a_n = a₁ × (r)ⁿ⁻¹

For the given sequence, the constant ratio is 21/10.5=2=

So, a₁₀ = 10.5 × (2)⁹

=10.5 × 512

= 5376 

5. Connect Geometric sequences and Exponential functions 

The exponential function can be written as a geometric sequence with the first term and constant ratio using the explicit formula. 

6. Connect Geometric sequences and Exponential functions 

The sum of the terms of a geometric sequence is a Geometric series. 

Let S_n be the sum of a geometric sequence with n terms. 

Exercise

• The constant ratio of the geometric sequence 3/5,3/2,15/4,75/8,… is .
• Write the recursive formula for a geometric sequence 2, 16, 128, 1024, …
• What is the 10th term of the geometric sequence 10.5, 21, 42, 84…?
• The first term of the sequence a_3=8(1/2)^7 is _.
  The constant ratio of the geometric sequence 10.5, 21, 42, 84… is __.

What we have learned

• A sequence in which the common difference between two consecutive terms is constant is called an arithmetic sequence.
• A sequence in which the constant ratio between two consecutive terms is constant is called a geometric sequence.

Concept Map