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# Geometric Sequences

Grade 10
Sep 14, 2022

## 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 an = r(an−1)

So, the recursive formula for the sequence 2, 10, 50, 250, … is an = 5(an−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 10th term of the geometric sequence 10.5, 21, 42, 84…?

Sol: Using the explicit formula, an = 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 Sn 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

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