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Question 7: 421924

The same sequence may be defined in three ways:

1. [tex]f(x)=3x+4[/tex] for [tex]x=\{1,2,3,\ldots\}[/tex]
2. [tex]a_1=4[/tex] and [tex]a_{n+1}=a_n+3[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex]
3. [tex]a_n=4+3(n-1)[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex]

They are all supposed to generate the same values, but one version may contain an error. If one version contains an error, identify the one with the error and fix it.

A. None of these have errors, they all generate the same sequence.
B. [tex]f(x)=3x+4[/tex] for [tex]x=\{1,2,3,\ldots\}[/tex]
It should say [tex]f(x)=3x+1[/tex] for [tex]x=\{1,2,3,\ldots\}[/tex].
C. [tex]a_n=4+3(n-1)[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex]
It should say [tex]a_n=3+4(n-1)[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex].
D. [tex]a_1=4[/tex] and [tex]a_{n+1}=a_n+3[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex]
It should say [tex]a_1=3[/tex] and [tex]a_{n+1}=a_n+4[/tex] for [tex]n=\{1,2,3,\ldots\}[/tex].

Sagot :

GinnyAnsweridn
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].
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