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Sagot :
To determine which option represents the formula for the given arithmetic sequence [tex]\(-20, -16, -12, -8, -4, \ldots\)[/tex], we need to examine the pattern and properties of arithmetic sequences.
1. Identify the first term and common difference:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-20\)[/tex].
- The sequence progresses as follows:
[tex]\[ -20, -16, -12, -8, -4, \ldots \][/tex]
- To find the common difference ([tex]\(d\)[/tex]), we subtract the first term from the second term:
[tex]\[ d = -16 - (-20) = -16 + 20 = 4 \][/tex]
- Thus, the common difference is [tex]\(4\)[/tex].
2. General formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
- The [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
- Substituting the values we found:
[tex]\[ a_n = -20 + (n-1)(4) \][/tex]
3. Verify each option against this formula:
- Let's check each given function to see which one matches this form.
- Option 1: [tex]\(f(n) = -20 - 4(n-1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 - 4(1-1) = -20 - 4 \cdot 0 = -20 \][/tex]
- Plug in [tex]\(n=2\)[/tex]:
[tex]\[ f(2) = -20 - 4(2-1) = -20 - 4 \cdot 1 = -24 \][/tex]
- The second term [tex]\(-24\)[/tex] does not match the sequence.
- Option 2: [tex]\(f(n) = -20 + 4(n-1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 + 4(1-1) = -20 + 4 \cdot 0 = -20 \][/tex]
- Plug in [tex]\(n=2\)[/tex]:
[tex]\[ f(2) = -20 + 4(2-1) = -20 + 4 \cdot 1 = -16 \][/tex]
- The second term [tex]\(-16\)[/tex] matches the sequence.
- Option 3: [tex]\(f(n) = -20 - 4(n+1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 - 4(1+1) = -20 - 8 = -28 \][/tex]
- The first term [tex]\(-28\)[/tex] does not match the sequence.
- Option 4: [tex]\(f(n) = -20 + 4(n+1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 + 4(1+1) = -20 + 8 = -12 \][/tex]
- The first term [tex]\(-12\)[/tex] does not match the sequence.
Upon verification, Option 2 [tex]\(f(n) = -20 + 4(n-1)\)[/tex] matches the given arithmetic sequence correctly.
Therefore, the correct formula for the sequence is:
[tex]\[ \boxed{2} \][/tex]
1. Identify the first term and common difference:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-20\)[/tex].
- The sequence progresses as follows:
[tex]\[ -20, -16, -12, -8, -4, \ldots \][/tex]
- To find the common difference ([tex]\(d\)[/tex]), we subtract the first term from the second term:
[tex]\[ d = -16 - (-20) = -16 + 20 = 4 \][/tex]
- Thus, the common difference is [tex]\(4\)[/tex].
2. General formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
- The [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
- Substituting the values we found:
[tex]\[ a_n = -20 + (n-1)(4) \][/tex]
3. Verify each option against this formula:
- Let's check each given function to see which one matches this form.
- Option 1: [tex]\(f(n) = -20 - 4(n-1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 - 4(1-1) = -20 - 4 \cdot 0 = -20 \][/tex]
- Plug in [tex]\(n=2\)[/tex]:
[tex]\[ f(2) = -20 - 4(2-1) = -20 - 4 \cdot 1 = -24 \][/tex]
- The second term [tex]\(-24\)[/tex] does not match the sequence.
- Option 2: [tex]\(f(n) = -20 + 4(n-1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 + 4(1-1) = -20 + 4 \cdot 0 = -20 \][/tex]
- Plug in [tex]\(n=2\)[/tex]:
[tex]\[ f(2) = -20 + 4(2-1) = -20 + 4 \cdot 1 = -16 \][/tex]
- The second term [tex]\(-16\)[/tex] matches the sequence.
- Option 3: [tex]\(f(n) = -20 - 4(n+1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 - 4(1+1) = -20 - 8 = -28 \][/tex]
- The first term [tex]\(-28\)[/tex] does not match the sequence.
- Option 4: [tex]\(f(n) = -20 + 4(n+1)\)[/tex]
- Plug in [tex]\(n=1\)[/tex]:
[tex]\[ f(1) = -20 + 4(1+1) = -20 + 8 = -12 \][/tex]
- The first term [tex]\(-12\)[/tex] does not match the sequence.
Upon verification, Option 2 [tex]\(f(n) = -20 + 4(n-1)\)[/tex] matches the given arithmetic sequence correctly.
Therefore, the correct formula for the sequence is:
[tex]\[ \boxed{2} \][/tex]
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