To solve this problem, we need to analyze and verify the three given ways of defining a sequence to see if they produce the same sequence or if there is an error in one of them.
1. First Definition: [tex]\( f(x) = 3x + 4 \)[/tex] for [tex]\( x = \{1, 2, 3, \ldots\} \)[/tex]:
- Let's generate the first few terms of the sequence:
- When [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 3(1) + 4 = 7 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 3(2) + 4 = 10 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 3(3) + 4 = 13 \)[/tex].
- And so on. The sequence generated is [tex]\( 7, 10, 13, \ldots \)[/tex].
2. Second Definition: [tex]\( a_1 = 4 \)[/tex] and [tex]\( a_{n+1} = a_n + 3 \)[/tex] for [tex]\( n = \{1, 2, 3, \ldots\} \)[/tex]:
- Let's generate the first few terms using the recursive definition:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_2 = a_1 + 3 = 4 + 3 = 7 \)[/tex]
- [tex]\( a_3 = a_2 + 3 = 7 + 3 = 10 \)[/tex]
- [tex]\( a_4 = a_3 + 3 = 10 + 3 = 13 \)[/tex]
- And so on. The sequence generated is [tex]\( 4, 7, 10, 13, 16, \ldots \)[/tex].
3. Third Definition: [tex]\( a_n = 4 + 3(n-1) \)[/tex] for [tex]\( n = \{1, 2, 3, \ldots\} \)[/tex]:
- Let's generate the first few terms using the explicit formula:
- When [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 4 + 3(1-1) = 4 + 3(0) = 4 \)[/tex].
- When [tex]\( n = 2 \)[/tex], [tex]\( a_2 = 4 + 3(2-1) = 4 + 3(1) = 7 \)[/tex].
- When [tex]\( n = 3 \)[/tex], [tex]\( a_3 = 4 + 3(3-1) = 4 + 3(2) = 10 \)[/tex].
- When [tex]\( n = 4 \)[/tex], [tex]\( a_4 = 4 + 3(4-1) = 4 + 3(3) = 13 \)[/tex].
- And so on. The sequence generated is [tex]\( 4, 7, 10, 13, \ldots \)[/tex].
After generating these sequences, we compare them:
1. The first definition generates the sequence [tex]\( 7, 10, 13, \ldots \)[/tex].
2. The second and third definitions both generate the sequence [tex]\( 4, 7, 10, 13, \ldots \)[/tex].
It's evident that the first definition does not match the other two. Therefore, there's an error in the first definition.
Fixing the Error:
- The function [tex]\( f(x) = 3x + 4 \)[/tex] should be adjusted to match the sequences generated by the second and third definitions.
- The correct function appears to be [tex]\( f(x) = 3x + 1 \)[/tex] instead of [tex]\( f(x) = 3x + 4 \)[/tex].
Thus, the correct answer is:
B [tex]\( f(x)=3 x+4 \)[/tex] for [tex]\( x=\{1,2,3, \ldots\} \)[/tex] It should say [tex]\( f(x)=3 x+1 \)[/tex] for [tex]\( x=\{1,2,3, \ldots\} \)[/tex].
