Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's find the explicit formula for the given arithmetic sequence: 9.2, 7.4, 5.6, 3.8, 2. We'll derive the first term and the common difference and then fit it into the standard form of an arithmetic sequence.
The sequence appears as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline n & 1 & 2 & 3 & 4 & 5 \\ \hline a_n & 9.2 & 7.4 & 5.6 & 3.8 & 2 \\ \hline \end{array} \][/tex]
First, we identify the elements needed to form the explicit formula for the sequence:
1. The first term [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = 9.2 \][/tex]
2. The common difference [tex]\(d\)[/tex], which is the difference between consecutive terms:
[tex]\[ d = a_2 - a_1 = 7.4 - 9.2 = -1.8 \][/tex]
The explicit formula for an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Plugging in the values we've identified:
[tex]\[ a_n = 9.2 + (n-1)(-1.8) \][/tex]
Let's elaborate on the formula provided in each option and verify them:
1. Option 1: [tex]\(a_n = 1 + 1.8(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 1 + 1.8(1-1) = 1 + 0 = 1 \][/tex]
- The first term should be 9.2, so this option is incorrect.
2. Option 2: [tex]\(a_n = 2 + 1.8(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 2 + 1.8(1-1) = 2 + 0 = 2 \][/tex]
- Again, the first term should be 9.2, so this option is incorrect.
3. Option 3: [tex]\(a_n = 9.2 + (-1.8)(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2 \][/tex]
- Verifying for [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11.0 \][/tex]
- This does not match the second term of the sequence, so this option is incorrect.
4. Option 4: [tex]\(a_n = 9.2 + (-1.8)(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2 \][/tex]
- Substituting [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4 \][/tex]
- Substituting [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6 \][/tex]
- Substituting [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8 \][/tex]
- Substituting [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2.0 \][/tex]
Thus, Option 4 correctly describes the explicit formula for the sequence.
Let's summarize:
The correct explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n-1) \][/tex]
So, the correct option is:
[tex]\[ a_n = 9.2 + (-1.8)(n-1) \][/tex]
The sequence appears as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline n & 1 & 2 & 3 & 4 & 5 \\ \hline a_n & 9.2 & 7.4 & 5.6 & 3.8 & 2 \\ \hline \end{array} \][/tex]
First, we identify the elements needed to form the explicit formula for the sequence:
1. The first term [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = 9.2 \][/tex]
2. The common difference [tex]\(d\)[/tex], which is the difference between consecutive terms:
[tex]\[ d = a_2 - a_1 = 7.4 - 9.2 = -1.8 \][/tex]
The explicit formula for an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Plugging in the values we've identified:
[tex]\[ a_n = 9.2 + (n-1)(-1.8) \][/tex]
Let's elaborate on the formula provided in each option and verify them:
1. Option 1: [tex]\(a_n = 1 + 1.8(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 1 + 1.8(1-1) = 1 + 0 = 1 \][/tex]
- The first term should be 9.2, so this option is incorrect.
2. Option 2: [tex]\(a_n = 2 + 1.8(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 2 + 1.8(1-1) = 2 + 0 = 2 \][/tex]
- Again, the first term should be 9.2, so this option is incorrect.
3. Option 3: [tex]\(a_n = 9.2 + (-1.8)(1-n)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2 \][/tex]
- Verifying for [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11.0 \][/tex]
- This does not match the second term of the sequence, so this option is incorrect.
4. Option 4: [tex]\(a_n = 9.2 + (-1.8)(n-1)\)[/tex]
- Substituting [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1-1) = 9.2 + 0 = 9.2 \][/tex]
- Substituting [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4 \][/tex]
- Substituting [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6 \][/tex]
- Substituting [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8 \][/tex]
- Substituting [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2.0 \][/tex]
Thus, Option 4 correctly describes the explicit formula for the sequence.
Let's summarize:
The correct explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n-1) \][/tex]
So, the correct option is:
[tex]\[ a_n = 9.2 + (-1.8)(n-1) \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
