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.
![Arithmetic sequence Geometric Sequences](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3469.png)
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.
![The common difference of the given sequence is +5](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3471.png)
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
![The common ratio of terms of the sequence is 2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3470.png)
3. Recursive formula for a sequence
- We can use the recursive formula to find the next term of a geometric sequence.
![Recursive formula for a sequence](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3472.png)
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.
![Explicit formula for a sequence](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3473.png)
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.
![Connect Geometric sequences and Exponential functions](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3474-1024x338.png)
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.
![Connect Geometric sequences and Exponential functions](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3475-1024x374.png)
![example](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2390.png)
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
![Concept Map:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3478.png)
![Concept Map:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-3477-1024x532.png)
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