Sure! Let's analyze each sequence step-by-step to determine the next three numbers and explain how these numbers were determined.
### Sequence A: 4, 7, 10, 13, 16, ...
#### (a) Next three numbers
This sequence is an arithmetic progression (AP) where the difference between consecutive terms is constant.
The common difference \(\( d \)\) can be found by subtracting any term from the next term:
\(\[ d = 7 - 4 = 3 \]\)
To find the next three terms:
1. \(\( 16 + 3 = 19 \)\)
2. \(\( 19 + 3 = 22 \)\)
3. \(\( 22 + 3 = 25 \)\)
So, the next three numbers are:
- 19
- 22
- 25
#### Sequence A:
(1) 19
(2) 22
(3) 25
### Sequence B: 5, 10, 20, 40, 80, ...
#### (a) Next three numbers
This sequence is a geometric progression (GP) where each term after the first is multiplied by a constant ratio.
The common ratio \(\( r \)\) can be found by dividing any term by the previous term:
\(\[ r = \frac{10}{5} = 2 \]\)
To find the next three terms:
1. \(\( 80 \times 2 = 160 \)\)
2. \(\( 160 \times 2 = 320 \)\)
3. \(\( 320 \times 2 = 640 \)\)
So, the next three numbers are:
- 160
- 320
- 640
#### Sequence B:
(1) 160
(2) 320
(3) 640
### Sequence C: 2, 5, 10, 17, 26, ...
#### (a) Next three numbers
This sequence follows a pattern where each term appears to be \(\( n^2 + 1 \)\) for \(\( n \)\).
For example:
1. \(\( 1^2 + 1 = 2 \)\)
2. \(\( 2^2 + 1 = 5 \)\)
3. \(\( 3^2 + 1 = 10 \)\)
4. \(\( 4^2 + 1 = 17 \)\)
5. \(\( 5^2 + 1 = 26 \)\)
To find the next three terms, we continue this pattern for \(\( n = 6, 7, 8 \)\):
1. \(\( 6^2 + 1 = 37 \)\)
2. \(\( 7^2 + 1 = 50 \)\)
3. \(\( 8^2 + 1 = 65 \)\)
So, the next three numbers are:
- 37
- 50
- 65
#### Sequence C:
(1) 37
(2) 50
(3) 65
### (b) How the next numbers were determined:
#### Sequence A:
This sequence is an arithmetic progression with a common difference of 3. Each term is obtained by adding 3 to the previous term.
#### Sequence B:
This sequence is a geometric progression with a common ratio of 2. Each term is obtained by multiplying the previous term by 2.
#### Sequence C:
This sequence follows the pattern \(\( n^2 + 1 \)\). Each term is calculated by taking the position number, squaring it, and adding 1.
If you have any more questions or need further clarification, feel free to ask!
