Let's analyze each statement step-by-step:
1. It can be represented using the formula [tex]\( f(x+1) = \frac{6}{5} f(x) \)[/tex] when [tex]\( f(1) = \frac{10}{3} \)[/tex]:
- Given [tex]\( f(1) = \frac{10}{3} \)[/tex], we use the formula [tex]\( f(x+1) = \frac{6}{5} f(x) \)[/tex] to find the next terms.
- For [tex]\( x = 1 \)[/tex], [tex]\( f(2) = \frac{6}{5} \cdot \frac{10}{3} = 4 \)[/tex].
- For [tex]\( x = 2 \)[/tex], [tex]\( f(3) = \frac{6}{5} \cdot 4 = \frac{24}{5} \)[/tex].
- For [tex]\( x = 3 \)[/tex], [tex]\( f(4) = \frac{6}{5} \cdot \frac{24}{5} = \frac{144}{25} \)[/tex].
- This matches the given sequence exactly.
- Therefore, this statement is true.
2. It can be represented using the formula [tex]\( f(x) = 4 \left( \frac{6}{5} \right)^x \)[/tex]:
- Let’s test this formula for the first term:
[tex]\( f(1) = 4 \left( \frac{6}{5} \right)^1 = \frac{24}{5} \)[/tex], which is not [tex]\(\frac{10}{3}\)[/tex].
- Since the first term does not match, this representation is false.
3. It can be represented using the formula [tex]\( f(x) = \frac{10}{3} \left( \frac{6}{5} \right)^{x-1} \)[/tex]:
- Let’s test this formula for a few terms:
- For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = \frac{10}{3} \left( \frac{6}{5} \right)^0 = \frac{10}{3} \)[/tex],
- For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = \frac{10}{3} \left( \frac{6}{5} \right)^1 = 4 \)[/tex],
- For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = \frac{10}{3} \left( \frac{6}{5} \right)^2 = \frac{24}{5} \)[/tex],
- For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = \frac{10}{3} \left( \frac{6}{5} \right)^3 = \frac{144}{25} \)[/tex].
- This matches the given sequence exactly.
- Therefore, this statement is true.
4. The domain of the sequence is all real numbers:
- The sequence is typically defined for integer values of [tex]\( x \)[/tex].
- In general, sequences are defined for natural numbers (positive integers).
- Therefore, this statement is false.
5. The range of the sequence is all natural numbers:
- The terms of the sequence are fractions ([tex]\(\frac{10}{3}, 4, \frac{24}{5}, \frac{144}{25}\)[/tex]).
- These are not all natural numbers.
- Therefore, this statement is false.
So, the statements that are true are:
1. It can be represented using the formula [tex]\( f(x+1) = \frac{6}{5} f(x) \)[/tex] when [tex]\( f(1) = \frac{10}{3} \)[/tex].
3. It can be represented using the formula [tex]\( f(x) = \frac{10}{3} \left( \frac{6}{5} \right)^{x-1} \)[/tex].
This leads to the result: [tex]\( ( \text{True, False, True, False, False} ) \)[/tex].
