Answer :
Let's analyze the given sequence: [tex]\( 5, 9, 13, 17, 21, \ldots \)[/tex].
### Step-by-Step Analysis:
1. Identify the Sequence Pattern:
Calculate the differences between consecutive terms:
[tex]\[ \begin{align*} 9 - 5 &= 4, \\ 13 - 9 &= 4, \\ 17 - 13 &= 4, \\ 21 - 17 &= 4. \end{align*} \][/tex]
The differences are constant, indicating that this is an arithmetic sequence with a common difference of [tex]\( 4 \)[/tex].
2. Find the General Formula:
The general form of an arithmetic sequence is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
Where [tex]\( a \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number.
For the given sequence:
- The first term [tex]\( a \)[/tex] is [tex]\( 5 \)[/tex].
- The common difference [tex]\( d \)[/tex] is [tex]\( 4 \)[/tex].
Substituting these values into the formula, we get:
[tex]\[ a_n = 5 + (n-1) \cdot 4 \][/tex]
Simplifying the expression:
[tex]\[ a_n = 5 + 4n - 4 = 4n + 1 \][/tex]
### Checking Against the Given Options:
1. Option 1: [tex]\( 5n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5n \][/tex]
This won't generate the correct sequence, as:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 0 \)[/tex]: which starts with 0, not 5.
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 5 \)[/tex]: which matches.
- For [tex]\( n = 2 \)[/tex], [tex]\( a_2 = 10 \)[/tex]: which does not match the sequence.
This option is incorrect.
2. Option 2: [tex]\( 5 + n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5 + n \][/tex]
This won't generate the correct sequence, as:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 5 \)[/tex]: which matches.
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 6 \)[/tex]: which does not match the sequence.
This option is incorrect.
3. Option 3: [tex]\( 5 + 4n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3 \)[/tex]
[tex]\[ a_n = 5 + 4n \][/tex]
This generates the correct sequence:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 5 \)[/tex] (matches).
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 9 \)[/tex] (matches).
- For [tex]\( n = 2 \)[/tex], [tex]\( a_2 = 13 \)[/tex] (matches).
- For [tex]\( n = 3 \)[/tex], [tex]\( a_3 = 17 \)[/tex] (matches).
- For [tex]\( n = 4 \)[/tex], [tex]\( a_4 = 21 \)[/tex] (matches).
This option is correct.
4. Option 4: [tex]\( 5h + 4 \)[/tex], where [tex]\( h \)[/tex] is equal to [tex]\( 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5h + 4 \][/tex]
This won't generate the correct sequence, as [tex]\( h \)[/tex] starts from [tex]\( 1 \)[/tex]:
- For [tex]\( h = 1 \)[/tex], [tex]\( a_1 = 9 \)[/tex]: doesn't start with 5.
- For [tex]\( h = 2 \)[/tex], [tex]\( a_2 = 14 \)[/tex]: doesn't give the correct sequence.
This option is incorrect.
### Conclusion:
The correct rule for the sequence [tex]\( 5, 9, 13, 17, 21, \ldots \)[/tex] is:
[tex]\[ \boxed{5 + 4n}, \text{ where } n \text{ is equal to } 0, 1, 2, 3 \][/tex]
### Step-by-Step Analysis:
1. Identify the Sequence Pattern:
Calculate the differences between consecutive terms:
[tex]\[ \begin{align*} 9 - 5 &= 4, \\ 13 - 9 &= 4, \\ 17 - 13 &= 4, \\ 21 - 17 &= 4. \end{align*} \][/tex]
The differences are constant, indicating that this is an arithmetic sequence with a common difference of [tex]\( 4 \)[/tex].
2. Find the General Formula:
The general form of an arithmetic sequence is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
Where [tex]\( a \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number.
For the given sequence:
- The first term [tex]\( a \)[/tex] is [tex]\( 5 \)[/tex].
- The common difference [tex]\( d \)[/tex] is [tex]\( 4 \)[/tex].
Substituting these values into the formula, we get:
[tex]\[ a_n = 5 + (n-1) \cdot 4 \][/tex]
Simplifying the expression:
[tex]\[ a_n = 5 + 4n - 4 = 4n + 1 \][/tex]
### Checking Against the Given Options:
1. Option 1: [tex]\( 5n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5n \][/tex]
This won't generate the correct sequence, as:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 0 \)[/tex]: which starts with 0, not 5.
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 5 \)[/tex]: which matches.
- For [tex]\( n = 2 \)[/tex], [tex]\( a_2 = 10 \)[/tex]: which does not match the sequence.
This option is incorrect.
2. Option 2: [tex]\( 5 + n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5 + n \][/tex]
This won't generate the correct sequence, as:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 5 \)[/tex]: which matches.
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 6 \)[/tex]: which does not match the sequence.
This option is incorrect.
3. Option 3: [tex]\( 5 + 4n \)[/tex], where [tex]\( n \)[/tex] is equal to [tex]\( 0, 1, 2, 3 \)[/tex]
[tex]\[ a_n = 5 + 4n \][/tex]
This generates the correct sequence:
- For [tex]\( n = 0 \)[/tex], [tex]\( a_0 = 5 \)[/tex] (matches).
- For [tex]\( n = 1 \)[/tex], [tex]\( a_1 = 9 \)[/tex] (matches).
- For [tex]\( n = 2 \)[/tex], [tex]\( a_2 = 13 \)[/tex] (matches).
- For [tex]\( n = 3 \)[/tex], [tex]\( a_3 = 17 \)[/tex] (matches).
- For [tex]\( n = 4 \)[/tex], [tex]\( a_4 = 21 \)[/tex] (matches).
This option is correct.
4. Option 4: [tex]\( 5h + 4 \)[/tex], where [tex]\( h \)[/tex] is equal to [tex]\( 1, 2, 3, 4 \)[/tex]
[tex]\[ a_n = 5h + 4 \][/tex]
This won't generate the correct sequence, as [tex]\( h \)[/tex] starts from [tex]\( 1 \)[/tex]:
- For [tex]\( h = 1 \)[/tex], [tex]\( a_1 = 9 \)[/tex]: doesn't start with 5.
- For [tex]\( h = 2 \)[/tex], [tex]\( a_2 = 14 \)[/tex]: doesn't give the correct sequence.
This option is incorrect.
### Conclusion:
The correct rule for the sequence [tex]\( 5, 9, 13, 17, 21, \ldots \)[/tex] is:
[tex]\[ \boxed{5 + 4n}, \text{ where } n \text{ is equal to } 0, 1, 2, 3 \][/tex]
