Sure, let's go through the sequence and answer each question step-by-step:
### 1.1.1 Explain in your own words how you get the next term in the sequence.
To find the next term in the sequence, multiply the previous term by [tex]\(-2\)[/tex]. This pattern is consistent throughout the sequence. For example:
- Starting with [tex]\( -2 \)[/tex],
- The next term is [tex]\( -2 \times (-2) = 4 \)[/tex],
- The term after that is [tex]\( 4 \times (-2) = -8 \)[/tex],
- And so forth.
### 1.1.2 Write down the next three terms in the pattern.
Given the initial terms of the sequence are [tex]\(-2, 4, -8\)[/tex]:
1. The next term after [tex]\(-8\)[/tex] is [tex]\( -8 \times (-2) = 16 \)[/tex],
2. Following [tex]\( 16 \)[/tex] is [tex]\( 16 \times (-2) = -32 \)[/tex],
3. And following [tex]\(-32\)[/tex] is [tex]\( -32 \times (-2) = 64 \)[/tex].
So, the next three terms are [tex]\( 16, -32, \text{ and } 64 \)[/tex].
### 1.1.3 Determine the general rule of the sequence.
The general formula for the [tex]\( n \)[/tex]-th term [tex]\( T(n) \)[/tex] of this sequence can be expressed as:
[tex]\[ T(n) = -2 \times (-2)^{(n-1)} \][/tex]
This formula generates the terms of the sequence starting from [tex]\( n = 1 \)[/tex].
### 1.1.4 Determine the [tex]\(10^{\text{th}}\)[/tex] term of the sequence.
Using the general formula:
[tex]\[ T(10) = -2 \times (-2)^{(10-1)} \][/tex]
[tex]\[ T(10) = -2 \times (-2)^9 \][/tex]
[tex]\[ T(10) = -2 \times 512 \][/tex]
[tex]\[ T(10) = 1024 \][/tex]
So, the [tex]\(10^{\text{th}}\)[/tex] term of the sequence is [tex]\( 1024 \)[/tex].
### 1.1.5 Which term will be equal to [tex]\(-128\)[/tex]?
We need to find [tex]\( n \)[/tex] such that:
[tex]\[ T(n) = -128 \][/tex]
Using the general formula:
[tex]\[ -2 \times (-2)^{(n-1)} = -128 \][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ (-2)^{(n-1)} = 64 \][/tex]
Since [tex]\( 64 = (-2)^6 \)[/tex]:
[tex]\[ n-1 = 6 \][/tex]
[tex]\[ n = 7 \][/tex]
Therefore, the [tex]\( 7^{\text{th}} \)[/tex] term of the sequence is equal to [tex]\(-128\)[/tex].
Summarizing:
1.1.1. The next term is found by multiplying the previous term by [tex]\(-2\)[/tex].
1.1.2. The next three terms are [tex]\( 16, -32, \text{ and } 64 \)[/tex].
1.1.3. The general rule is [tex]\( T(n) = -2 \times (-2)^{(n-1)} \)[/tex].
1.1.4. The [tex]\( 10^{\text{th}} \)[/tex] term is [tex]\( 1024 \)[/tex].
1.1.5. The [tex]\(7^{\text{th}}\)[/tex] term is equal to [tex]\(-128\)[/tex].
