## 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} = a_{1} × (r)^{n−1}

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

So, a_{10} = 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

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