To determine which statement is true, we need to analyze the growth rates of both sequences for \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \) based on the provided values.
Values from the sequences:
-
For \( f(x) \):
- \( f(1) = 4 \)
- \( f(2) = 8 \)
- \( f(3) = 16 \)
- \( f(4) = 32 \)
- \( f(5) = 64 \)
- \( f(6) = 128 \)
- \( f(7) = 256 \)
- \( f(8) = 512 \)
- \( f(9) = 1024 \)
-
For \( g(x) \):
- \( g(1) = 12 \)
- \( g(2) = 22 \)
- \( g(3) = 32 \)
- \( g(4) = 42 \)
- \( g(5) = 52 \)
- \( g(6) = 62 \)
- \( g(7) = 72 \)
- \( g(8) = 82 \)
- \( g(9) = 92 \)
Determine the growth rates:
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Growth of \( f(x) \):
- \( f(1) \) to \( f(2) \): \( 8 - 4 = 4 \)
- \( f(2) \) to \( f(3) \): \( 16 - 8 = 8 \)
- \( f(3) \) to \( f(4) \): \( 32 - 16 = 16 \)
- \( f(4) \) to \( f(5) \): \( 64 - 32 = 32 \)
- \( f(5) \) to \( f(6) \): \( 128 - 64 = 64 \)
- \( f(6) \) to \( f(7) \): \( 256 - 128 = 128 \)
- \( f(7) \) to \( f(8) \): \( 512 - 256 = 256 \)
- \( f(8) \) to \( f(9) \): \( 1024 - 512 = 512 \)
-
Growth of \( g(x) \):
- \( g(1) \) to \( g(2) \): \( 22 - 12 = 10 \)
- \( g(2) \) to \( g(3) \): \( 32 - 22 = 10 \)
- \( g(3) \) to \( g(4) \): \( 42 - 32 = 10 \)
- \( g(4) \) to \( g(5) \): \( 52 - 42 = 10 \)
- \( g(5) \) to \( g(6) \): \( 62 - 52 = 10 \)
- \( g(6) \) to \( g(7) \): \( 72 - 62 = 10 \)
- \( g(7) \) to \( g(8) \): \( 82 - 72 = 10 \)
- \( g(8) \) to \( g(9) \): \( 92 - 82 = 10 \)
Analyze the statements:
-
Statement #1: While the growth rate of \( g(x) \) is initially greater than the growth rate of \( f(x) \), the growth rate of \( f(x) \) keeps increasing and eventually surpasses the growth rate of \( g(x) \).
- False: The initial growth of \( g(x) \) isn't greater; \( g(x) \) is constant at \( 10 \) and \( f(x) \) grows from \( 4 \) to much higher values.
-
Statement #2: The growth rate of \( g(x) \) is greater than the growth rate of \( f(x) \) between the second term and the fifth term.
- False: \( g(x) \)'s growth is constant at \( 10 \), while \( f(x) \) is increasing and surpasses \( g(x) \).
-
Statement #3: While the growth rate of \( f(x) \) is initially greater than the growth rate of \( g(x) \), the growth rate of \( g(x) \) keeps increasing and eventually surpasses the growth rate of \( f(x) \).
- False: \( g(x) \) has a constant growth, while \( f(x) \) outpaces it significantly.
Thus, the true statement is:
None of the statements are true.
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