7.2.3. Find the rule for the $$ n $$th term of the pattern (5) 7.2.4. Calculate the $$ 15^{ ext {th }} $$ term using your rule in 7.2.3. 7.3. Given the pattem $$ 2 ;-4 ; 8 ;-16 ; 32 ; \ldots ; $$ 7.3.1. Find the next term in the above pattern 7.3.2. Calculate the constant ratio

Question

7.2.3. Find the rule for the $$ n $$th term of the pattern (5) 7.2.4. Calculate the $$ 15^{	ext {th }} $$ term using your rule in 7.2.3. 7.3. Given the pattem $$ 2 ;-4 ; 8 ;-16 ; 32 ; \ldots ; $$ 7.3.1. Find the next term in the above pattern 7.3.2. Calculate the constant ratio

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### Problem 7.2.3: Find the rule for the $$ n $$th term of the pattern (5)

To find the $$ n $$th term of the pattern, we need to identify the sequence. However, the specific pattern is not provided in your message. Assuming the pattern is a simple arithmetic or geometric sequence, we can derive a general formula.

If the pattern is arithmetic, the $$ n $$th term can be expressed as:
$$
a_n = a_1 + (n-1)d
$$
where $$ a_1 $$ is the first term and $$ d $$ is the common difference.

If the pattern is geometric, the $$ n $$th term can be expressed as:
$$
a_n = a_1 \cdot r^{(n-1)}
$$
where $$ a_1 $$ is the first term and $$ r $$ is the common ratio.

Since the specific pattern is not provided, please clarify the sequence so I can provide the exact rule.

### Problem 7.2.4: Calculate the $$ 15^{\text{th}} $$ term using your rule in 7.2.3

Once we have the rule from 7.2.3, we can substitute $$ n = 15 $$ into the formula to find the $$ 15^{\text{th}} $$ term.

### Problem 7.3: Given the pattern $$ 2, -4, 8, -16, 32, \ldots $$

#### Problem 7.3.1: Find the next term in the above pattern

To find the next term, we can observe the pattern:
- The first term is $$ 2 $$.
- The second term is $$ -4 $$ (which is $$ 2 \times -2 $$).
- The third term is $$ 8 $$ (which is $$ -4 \times -2 $$).
- The fourth term is $$ -16 $$ (which is $$ 8 \times -2 $$).
- The fifth term is $$ 32 $$ (which is $$ -16 \times -2 $$).

From this, we can see that each term is multiplied by $$ -2 $$ to get the next term. Therefore, the next term after $$ 32 $$ will be:
$$
32 \times -2 = -64
$$

#### Problem 7.3.2: Calculate the constant ratio

The constant ratio $$ r $$ can be calculated as follows:
$$
r = \frac{\text{next term}}{\text{previous term}}
$$
Using the terms we have:
- From $$ 2 $$ to $$ -4 $$: $$ r = \frac{-4}{2} = -2 $$
- From $$ -4 $$ to $$ 8 $$: $$ r = \frac{8}{-4} = -2 $$
- From $$ 8 $$ to $$ -16 $$: $$ r = \frac{-16}{8} = -2 $$
- From $$ -16 $$ to $$ 32 $$: $$ r = \frac{32}{-16} = -2 $$

Thus, the constant ratio is $$ -2 $$.

### Summary of Results
- **7.2.3**: Please provide the specific pattern to find the $$ n $$th term rule.
- **7.2.4**: Can be calculated once the rule is provided.
- **7.3.1**: The next term is $$ -64 $$.
- **7.3.2**: The constant ratio is $$ -2 $$.
- **7.2.3**: Cannot determine the $$ n $$th term rule without the pattern. - **7.2.4**: Cannot calculate the $$ 15^{\text{th}} $$ term without the rule. - **7.3.1**: The next term is $$ -64 $$. - **7.3.2**: The constant ratio is $$ -2 $$.

7.2.3. Find the rule for the th term of the pattern (5)
7.2.4. Calculate the term using your rule in 7.2.3.
7.3. Given the pattem
7.3.1. Find the next term in the above pattern
7.3.2. Calculate the constant ratio

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Answer

Solution

Problem 7.2.3: Find the rule for the th term of the pattern (5)

Problem 7.2.4: Calculate the term using your rule in 7.2.3

Problem 7.3: Given the pattern

Problem 7.3.1: Find the next term in the above pattern

Problem 7.3.2: Calculate the constant ratio

Summary of Results

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7.2.3. Find the rule for the th term of the pattern (5) 7.2.4. Calculate the term using your rule in 7.2.3. 7.3. Given the pattem 7.3.1. Find the next term in the above pattern 7.3.2. Calculate the constant ratio