Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the explicit formula for the arithmetic sequence given in the table, let's follow a detailed, step-by-step process.
1. Identify Key Features of the Sequence:
- First few terms of the sequence are: 9.2, 7.4, 5.6, 3.8, 2.
- These terms decrease as [tex]\( n \)[/tex] increases, which suggests a negative common difference.
2. Calculate the Common Difference:
[tex]\[ d = a_2 - a_1 = 7.4 - 9.2 = -1.8 \][/tex]
The common difference [tex]\( d \)[/tex] is [tex]\(-1.8\)[/tex].
3. Identify the First Term:
[tex]\[ a_1 = 9.2 \][/tex]
4. Formulate the General Formula:
The general formula for an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \times d \][/tex]
Plugging in the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex], we get:
[tex]\[ a_n = 9.2 + (n - 1) \times (-1.8) \][/tex]
Simplifying, we get:
[tex]\[ a_n = 9.2 - 1.8(n - 1) = 9.2 - 1.8n + 1.8 = 11 - 1.8n \][/tex]
5. Evaluate the Given Options:
- Option a: [tex]\( a_n = 1 + 1.8(n - 1) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 1 + 1.8(1-1) = 1 \quad (\text{not } 9.2) \][/tex]
- Option b: [tex]\( a_n = 2 + 1.8(1 - n) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 2 + 1.8(1-1) = 2 \quad (\text{not } 9.2) \][/tex]
- Option c: [tex]\( a_n = 9.2 + (-1.8)(1 - n) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)} \][/tex]
[tex]\[ n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11 \quad (\text{not } 7.4) \][/tex]
- Option d: [tex]\( a_n = 9.2 + (-1.8)(n - 1) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)} \][/tex]
[tex]\[ n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4 \quad \text{(matches)} \][/tex]
[tex]\[ n = 3 \Rightarrow a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6 \quad \text{(matches)} \][/tex]
[tex]\[ n = 4 \Rightarrow a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8 \quad \text{(matches)} \][/tex]
[tex]\[ n = 5 \Rightarrow a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2 \quad \text{(matches)} \][/tex]
6. Determine the Correct Formula:
After evaluating the given options, we see that the only option that fits the first five terms of the sequence is the formula:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
Therefore, the explicit formula for the given sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
1. Identify Key Features of the Sequence:
- First few terms of the sequence are: 9.2, 7.4, 5.6, 3.8, 2.
- These terms decrease as [tex]\( n \)[/tex] increases, which suggests a negative common difference.
2. Calculate the Common Difference:
[tex]\[ d = a_2 - a_1 = 7.4 - 9.2 = -1.8 \][/tex]
The common difference [tex]\( d \)[/tex] is [tex]\(-1.8\)[/tex].
3. Identify the First Term:
[tex]\[ a_1 = 9.2 \][/tex]
4. Formulate the General Formula:
The general formula for an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \times d \][/tex]
Plugging in the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex], we get:
[tex]\[ a_n = 9.2 + (n - 1) \times (-1.8) \][/tex]
Simplifying, we get:
[tex]\[ a_n = 9.2 - 1.8(n - 1) = 9.2 - 1.8n + 1.8 = 11 - 1.8n \][/tex]
5. Evaluate the Given Options:
- Option a: [tex]\( a_n = 1 + 1.8(n - 1) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 1 + 1.8(1-1) = 1 \quad (\text{not } 9.2) \][/tex]
- Option b: [tex]\( a_n = 2 + 1.8(1 - n) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 2 + 1.8(1-1) = 2 \quad (\text{not } 9.2) \][/tex]
- Option c: [tex]\( a_n = 9.2 + (-1.8)(1 - n) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)} \][/tex]
[tex]\[ n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(1-2) = 9.2 + 1.8 = 11 \quad (\text{not } 7.4) \][/tex]
- Option d: [tex]\( a_n = 9.2 + (-1.8)(n - 1) \)[/tex]
[tex]\[ \text{For } n = 1, 2, 3, 4, 5: \][/tex]
[tex]\[ n = 1 \Rightarrow a_1 = 9.2 + (-1.8)(1-1) = 9.2 \quad \text{(matches)} \][/tex]
[tex]\[ n = 2 \Rightarrow a_2 = 9.2 + (-1.8)(2-1) = 9.2 - 1.8 = 7.4 \quad \text{(matches)} \][/tex]
[tex]\[ n = 3 \Rightarrow a_3 = 9.2 + (-1.8)(3-1) = 9.2 - 3.6 = 5.6 \quad \text{(matches)} \][/tex]
[tex]\[ n = 4 \Rightarrow a_4 = 9.2 + (-1.8)(4-1) = 9.2 - 5.4 = 3.8 \quad \text{(matches)} \][/tex]
[tex]\[ n = 5 \Rightarrow a_5 = 9.2 + (-1.8)(5-1) = 9.2 - 7.2 = 2 \quad \text{(matches)} \][/tex]
6. Determine the Correct Formula:
After evaluating the given options, we see that the only option that fits the first five terms of the sequence is the formula:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
Therefore, the explicit formula for the given sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
