Let's analyze each statement given the geometric sequence:
Statement 1: The domain is the set of natural numbers.
- In a geometric sequence, each term is defined based on a previous term and starts from an initial term. Typically, sequences start from the first term and proceed to the second, third, and so on. Hence, the domain of this sequence is indeed the set of natural numbers [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Conclusion: This statement is true.
Statement 2: The range is the set of natural numbers.
- A geometric sequence can be determined by a formula involving division or multiplication by a constant ratio. In this particular case, the constant ratio is [tex]\(\frac{3}{2}\)[/tex]. As the sequence progresses, the terms can become fractional rather than strictly natural numbers.
- Conclusion: This statement is false.
Statement 3: The recursive formula representing the sequence is [tex]\(f(x+1) = \frac{3}{2} f(x)\)[/tex] when [tex]\(f(1) = 4\)[/tex].
- The recursive formula of a geometric sequence gives the next term based on the current term multiplied by the common ratio. For this sequence:
- Common ratio ([tex]\(r\)[/tex]) = [tex]\(\frac{3}{2}\)[/tex]
- First term ([tex]\(f(1)\)[/tex]) = 4.
- Thus, it correctly describes the relationship [tex]\(f(x+1) = \frac{3}{2} f(x)\)[/tex] with the initial term [tex]\(f(1) = 4\)[/tex].
- Conclusion: This statement is true.
Statement 4: An explicit formula representing the sequence is [tex]\(f(x) = 4 \left(\frac{3}{2}\right)^x\)[/tex].
- The explicit formula for a geometric sequence generally is [tex]\(f(x) = f(1) \cdot r^{(x-1)}\)[/tex].
- Here, [tex]\(f(1) = 4\)[/tex] and [tex]\(r = \frac{3}{2}\)[/tex].
- So the correct formula would be [tex]\(f(x) = 4 \left(\frac{3}{2}\right)^{x-1}\)[/tex].
- Therefore, [tex]\(f(x) = 4 \left(\frac{3}{2}\right)^x\)[/tex] is incorrect because it doesn't account for the correct power of the ratio.
- Conclusion: This statement is false.
Statement 5: The sequence shows exponential growth.
- A geometric sequence where the common ratio [tex]\(r > 1\)[/tex] demonstrates exponential growth, as each term increases multiplicatively.
- Here, since the common ratio [tex]\(\frac{3}{2} > 1\)[/tex], the sequence indeed grows exponentially.
- Conclusion: This statement is true.
Summary of Evaluated Statements:
- The domain is the set of natural numbers. (True)
- The range is the set of natural numbers. (False)
- The recursive formula representing the sequence is [tex]\(f(x+1) = \frac{3}{2} f(x)\)[/tex] when [tex]\(f(1) = 4\)[/tex]. (True)
- An explicit formula representing the sequence is [tex]\(f(x) = 4 \left(\frac{3}{2}\right)^x\)[/tex]. (False)
- The sequence shows exponential growth. (True)
Thus, the true statements for the given geometric sequence are:
- The domain is the set of natural numbers.
- The recursive formula representing the sequence is [tex]\(f(x+1) = \frac{3}{2} f(x)\)[/tex] when [tex]\(f(1) = 4\)[/tex].
- The sequence shows exponential growth.
