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Consider the pattern [tex]$-2, 4, -8, \ldots$[/tex]

1. Explain in your own words how you get the next term. (1)

2. Write down the next three terms in the pattern. (3)

3. Determine the general rule of the sequence. (2)

4. Determine the [tex]$10^{\text{th}}$[/tex] term of the sequence. (3)

5. Which term will be equal to [tex]$-128$[/tex]?

Sagot :

GinnyAnsweridn
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].
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