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Sagot :
To determine which explicit formula represents the given arithmetic sequence, we need to calculate the terms of the sequence using each of the provided options and compare them to the given values of [tex]\( a_n \)[/tex].
Given the sequence values:
[tex]\[ n: 1, 2, 3, 4, 5 \][/tex]
[tex]\[ a_n: 9.2, 7.4, 5.6, 3.8, 2.0 \][/tex]
Let's evaluate each of the explicit formulas with these values of [tex]\( n \)[/tex].
### Option 1: [tex]\( a_n = 1 + 1.8(n - 1) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 1 + 1.8(1 - 1) = 1 + 0 = 1.0 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 1 + 1.8(2 - 1) = 1 + 1.8 = 2.8 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 1 + 1.8(3 - 1) = 1 + 3.6 = 4.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 1 + 1.8(4 - 1) = 1 + 5.4 = 6.4 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 1 + 1.8(5 - 1) = 1 + 7.2 = 8.2 \][/tex]
The sequence generated by this option is [tex]\([1.0, 2.8, 4.6, 6.4, 8.2]\)[/tex], which does not match our given sequence.
### Option 2: [tex]\( a_n = 2 + 1.8(1 - n) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 2 + 1.8(1 - 1) = 2 + 0 = 2.0 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 + 1.8(1 - 2) = 2 - 1.8 = 0.2 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 2 + 1.8(1 - 3) = 2 - 3.6 = -1.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 2 + 1.8(1 - 4) = 2 - 5.4 = -3.4 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 2 + 1.8(1 - 5) = 2 - 7.2 = -5.2 \][/tex]
The sequence generated by this option is [tex]\([2.0, 0.2, -1.6, -3.4, -5.2]\)[/tex], which does not match our given sequence.
### Option 3: [tex]\( a_n = 9.2 + (-1.8)(1 - n) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1 - 1) = 9.2 + 0 = 9.2 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(1 - 2) = 9.2 + 1.8 = 11.0 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(1 - 3) = 9.2 + 3.6 = 12.8 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(1 - 4) = 9.2 + 5.4 = 14.6 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(1 - 5) = 9.2 + 7.2 = 16.4 \][/tex]
The sequence generated by this option is [tex]\([9.2, 11.0, 12.8, 14.6, 16.4]\)[/tex], which does not match our given sequence.
### Option 4: [tex]\( a_n = 9.2 + (-1.8)(n - 1) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1 - 1) = 9.2 + 0 = 9.2 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(2 - 1) = 9.2 - 1.8 = 7.4 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(3 - 1) = 9.2 - 3.6 = 5.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(4 - 1) = 9.2 - 5.4 = 3.8 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(5 - 1) = 9.2 - 7.2 = 2.0 \][/tex]
The sequence generated by this option is [tex]\([9.2, 7.4, 5.6, 3.8, 2.0]\)[/tex], which matches our given sequence exactly.
Thus, the explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
Given the sequence values:
[tex]\[ n: 1, 2, 3, 4, 5 \][/tex]
[tex]\[ a_n: 9.2, 7.4, 5.6, 3.8, 2.0 \][/tex]
Let's evaluate each of the explicit formulas with these values of [tex]\( n \)[/tex].
### Option 1: [tex]\( a_n = 1 + 1.8(n - 1) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 1 + 1.8(1 - 1) = 1 + 0 = 1.0 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 1 + 1.8(2 - 1) = 1 + 1.8 = 2.8 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 1 + 1.8(3 - 1) = 1 + 3.6 = 4.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 1 + 1.8(4 - 1) = 1 + 5.4 = 6.4 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 1 + 1.8(5 - 1) = 1 + 7.2 = 8.2 \][/tex]
The sequence generated by this option is [tex]\([1.0, 2.8, 4.6, 6.4, 8.2]\)[/tex], which does not match our given sequence.
### Option 2: [tex]\( a_n = 2 + 1.8(1 - n) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 2 + 1.8(1 - 1) = 2 + 0 = 2.0 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 + 1.8(1 - 2) = 2 - 1.8 = 0.2 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 2 + 1.8(1 - 3) = 2 - 3.6 = -1.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 2 + 1.8(1 - 4) = 2 - 5.4 = -3.4 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 2 + 1.8(1 - 5) = 2 - 7.2 = -5.2 \][/tex]
The sequence generated by this option is [tex]\([2.0, 0.2, -1.6, -3.4, -5.2]\)[/tex], which does not match our given sequence.
### Option 3: [tex]\( a_n = 9.2 + (-1.8)(1 - n) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1 - 1) = 9.2 + 0 = 9.2 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(1 - 2) = 9.2 + 1.8 = 11.0 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(1 - 3) = 9.2 + 3.6 = 12.8 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(1 - 4) = 9.2 + 5.4 = 14.6 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(1 - 5) = 9.2 + 7.2 = 16.4 \][/tex]
The sequence generated by this option is [tex]\([9.2, 11.0, 12.8, 14.6, 16.4]\)[/tex], which does not match our given sequence.
### Option 4: [tex]\( a_n = 9.2 + (-1.8)(n - 1) \)[/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 9.2 + (-1.8)(1 - 1) = 9.2 + 0 = 9.2 \][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 9.2 + (-1.8)(2 - 1) = 9.2 - 1.8 = 7.4 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 9.2 + (-1.8)(3 - 1) = 9.2 - 3.6 = 5.6 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 9.2 + (-1.8)(4 - 1) = 9.2 - 5.4 = 3.8 \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 9.2 + (-1.8)(5 - 1) = 9.2 - 7.2 = 2.0 \][/tex]
The sequence generated by this option is [tex]\([9.2, 7.4, 5.6, 3.8, 2.0]\)[/tex], which matches our given sequence exactly.
Thus, the explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 9.2 + (-1.8)(n - 1) \][/tex]
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