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To determine the correct rule for the [tex]\(n\)[/tex]th term of an arithmetic sequence where [tex]\(a_{10} = 46\)[/tex] and the common difference [tex]\(d = 3\)[/tex], we can follow these steps:
1. Recall the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Here, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] represents the term number.
2. Use given values to find the first term [tex]\(a_1\)[/tex]:
We know that:
[tex]\[ a_{10} = 46 \][/tex]
Substituting [tex]\(n = 10\)[/tex] and [tex]\(d = 3\)[/tex] into the formula for [tex]\(a_n\)[/tex]:
[tex]\[ a_{10} = a_1 + (10-1) \cdot 3 \][/tex]
This simplifies to:
[tex]\[ 46 = a_1 + 9 \cdot 3 \][/tex]
[tex]\[ 46 = a_1 + 27 \][/tex]
Solving for [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = 46 - 27 \][/tex]
[tex]\[ a_1 = 19 \][/tex]
3. Determine the candidate formulas using the found first term [tex]\(a_1 = 19\)[/tex]:
4. Test each given formula to see which one matches the sequence:
We have four candidate formulas:
[tex]\[ a_n = 43 + 3n \][/tex]
[tex]\[ a_n = 43 - 3n \][/tex]
[tex]\[ a_n = 16 + 3n \][/tex]
[tex]\[ a_n = 16 - 3n \][/tex]
Let's test each formula:
- For [tex]\(a_n = 43 + 3n\)[/tex]:
[tex]\[ a_{10} = 43 + 3 \cdot 10 = 43 + 30 = 73 \][/tex]
This is not equal to 46, so this formula is incorrect.
- For [tex]\(a_n = 43 - 3n\)[/tex]:
[tex]\[ a_{10} = 43 - 3 \cdot 10 = 43 - 30 = 13 \][/tex]
This is not equal to 46, so this formula is also incorrect.
- For [tex]\(a_n = 16 + 3n\)[/tex]:
[tex]\[ a_{10} = 16 + 3 \cdot 10 = 16 + 30 = 46 \][/tex]
This matches the given [tex]\(a_{10}\)[/tex], indicating this formula is correct.
- For [tex]\(a_n = 16 - 3n\)[/tex]:
[tex]\[ a_{10} = 16 - 3 \cdot 10 = 16 - 30 = -14 \][/tex]
This is not equal to 46, so this formula is incorrect.
Thus, the correct rule for the [tex]\(n\)[/tex]th term of the sequence is:
[tex]\[ a_n = 16 + 3n \][/tex]
1. Recall the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Here, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] represents the term number.
2. Use given values to find the first term [tex]\(a_1\)[/tex]:
We know that:
[tex]\[ a_{10} = 46 \][/tex]
Substituting [tex]\(n = 10\)[/tex] and [tex]\(d = 3\)[/tex] into the formula for [tex]\(a_n\)[/tex]:
[tex]\[ a_{10} = a_1 + (10-1) \cdot 3 \][/tex]
This simplifies to:
[tex]\[ 46 = a_1 + 9 \cdot 3 \][/tex]
[tex]\[ 46 = a_1 + 27 \][/tex]
Solving for [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = 46 - 27 \][/tex]
[tex]\[ a_1 = 19 \][/tex]
3. Determine the candidate formulas using the found first term [tex]\(a_1 = 19\)[/tex]:
4. Test each given formula to see which one matches the sequence:
We have four candidate formulas:
[tex]\[ a_n = 43 + 3n \][/tex]
[tex]\[ a_n = 43 - 3n \][/tex]
[tex]\[ a_n = 16 + 3n \][/tex]
[tex]\[ a_n = 16 - 3n \][/tex]
Let's test each formula:
- For [tex]\(a_n = 43 + 3n\)[/tex]:
[tex]\[ a_{10} = 43 + 3 \cdot 10 = 43 + 30 = 73 \][/tex]
This is not equal to 46, so this formula is incorrect.
- For [tex]\(a_n = 43 - 3n\)[/tex]:
[tex]\[ a_{10} = 43 - 3 \cdot 10 = 43 - 30 = 13 \][/tex]
This is not equal to 46, so this formula is also incorrect.
- For [tex]\(a_n = 16 + 3n\)[/tex]:
[tex]\[ a_{10} = 16 + 3 \cdot 10 = 16 + 30 = 46 \][/tex]
This matches the given [tex]\(a_{10}\)[/tex], indicating this formula is correct.
- For [tex]\(a_n = 16 - 3n\)[/tex]:
[tex]\[ a_{10} = 16 - 3 \cdot 10 = 16 - 30 = -14 \][/tex]
This is not equal to 46, so this formula is incorrect.
Thus, the correct rule for the [tex]\(n\)[/tex]th term of the sequence is:
[tex]\[ a_n = 16 + 3n \][/tex]
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