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To determine which recursive formula represents the same arithmetic sequence as the given explicit formula [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex], let's break down the given explicit formula and compare it with each of the recursive options.
### Step-by-Step Solution
1. Identify the first term from the explicit formula.
- The explicit formula is [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex].
- For the first term when [tex]\( n=1 \)[/tex]:
[tex]\[ a_1 = 5 + (1-1) \cdot 2 = 5 + 0 = 5 \][/tex]
- Thus, the first term [tex]\( a_1 \)[/tex] is 5.
2. Identify the common difference.
- The explicit formula is written in the form [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex].
- Comparing [tex]\( a_n = 5 + (n-1)\cdot 2 \)[/tex] to this form, the common difference [tex]\( d \)[/tex] is 2.
3. Compare with each recursive formula:
- Option A:
[tex]\[ \left\{ \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} + 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 2, which does not match our identified first term 5.
- The common difference is 5, which does not match our identified common difference 2.
- Thus, this option is incorrect.
- Option B:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = a_{n-1} + 2 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 5, which matches our identified first term.
- The common difference is 2, which matches our identified common difference.
- Thus, this option matches the explicit formula.
- Option C:
[tex]\[ \left\{ \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} \cdot 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 2, which does not match the identified first term 5.
- Additionally, multiplying by 5 does not represent the arithmetic sequence with a common difference 2.
- Thus, this option is incorrect.
- Option D:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = (a_{n-1} + 2) \cdot 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 5, which matches our identified first term.
- However, the expression [tex]\( a_n = (a_{n-1} + 2) \cdot 5 \)[/tex] does not represent the arithmetic sequence [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex]. It represents a different type of sequence entirely.
- Thus, this option is incorrect.
4. Conclusion:
- The correct recursive formula that represents the same arithmetic sequence as the explicit formula [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex] is:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = a_{n-1} + 2 \end{array} \right. \][/tex]
- Therefore, the correct choice is Option B.
### Step-by-Step Solution
1. Identify the first term from the explicit formula.
- The explicit formula is [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex].
- For the first term when [tex]\( n=1 \)[/tex]:
[tex]\[ a_1 = 5 + (1-1) \cdot 2 = 5 + 0 = 5 \][/tex]
- Thus, the first term [tex]\( a_1 \)[/tex] is 5.
2. Identify the common difference.
- The explicit formula is written in the form [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex].
- Comparing [tex]\( a_n = 5 + (n-1)\cdot 2 \)[/tex] to this form, the common difference [tex]\( d \)[/tex] is 2.
3. Compare with each recursive formula:
- Option A:
[tex]\[ \left\{ \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} + 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 2, which does not match our identified first term 5.
- The common difference is 5, which does not match our identified common difference 2.
- Thus, this option is incorrect.
- Option B:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = a_{n-1} + 2 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 5, which matches our identified first term.
- The common difference is 2, which matches our identified common difference.
- Thus, this option matches the explicit formula.
- Option C:
[tex]\[ \left\{ \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} \cdot 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 2, which does not match the identified first term 5.
- Additionally, multiplying by 5 does not represent the arithmetic sequence with a common difference 2.
- Thus, this option is incorrect.
- Option D:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = (a_{n-1} + 2) \cdot 5 \end{array} \right. \][/tex]
- Here, the first term [tex]\( a_1 \)[/tex] is 5, which matches our identified first term.
- However, the expression [tex]\( a_n = (a_{n-1} + 2) \cdot 5 \)[/tex] does not represent the arithmetic sequence [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex]. It represents a different type of sequence entirely.
- Thus, this option is incorrect.
4. Conclusion:
- The correct recursive formula that represents the same arithmetic sequence as the explicit formula [tex]\( a_n = 5 + (n-1) \cdot 2 \)[/tex] is:
[tex]\[ \left\{ \begin{array}{l} a_1 = 5 \\ a_n = a_{n-1} + 2 \end{array} \right. \][/tex]
- Therefore, the correct choice is Option B.
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