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1. Using the General Formula of an Arithmetic Sequence:
For an arithmetic sequence, the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) can be expressed as:
[tex]\[ a_n = a_1 + (n-1) d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
2. Getting Information from the Problem Statements:
- The sixth term ([tex]\(a_6\)[/tex]) is 3 times the fourth term ([tex]\(a_4\)[/tex]):
[tex]\[ a_6 = 3 \cdot a_4 \][/tex]
- The sum of the first three terms ([tex]\(a_1 + a_2 + a_3\)[/tex]) is -12:
[tex]\[ a_1 + a_2 + a_3 = -12 \][/tex]
3. Expressing [tex]\(a_6\)[/tex] and [tex]\(a_4\)[/tex] in Terms of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
Using the formula [tex]\(a_n = a_1 + (n-1) d\)[/tex]:
- The fourth term:
[tex]\[ a_4 = a_1 + 3d \][/tex]
- The sixth term:
[tex]\[ a_6 = a_1 + 5d \][/tex]
Substituting these into the given relation [tex]\(a_6 = 3 \cdot a_4\)[/tex]:
[tex]\[ a_1 + 5d = 3(a_1 + 3d) \][/tex]
4. Solving for [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
Expanding the equation we get:
[tex]\[ a_1 + 5d = 3a_1 + 9d \][/tex]
Rearranging to find [tex]\(a_1\)[/tex]:
[tex]\[ 5d - 9d = 3a_1 - a_1 \][/tex]
[tex]\[ -4d = 2a_1 \][/tex]
[tex]\[ a_1 = -2d \][/tex]
5. Using Sum of the First Three Terms:
The formula for the sum of the first [tex]\(n\)[/tex] terms [tex]\(S_n\)[/tex] of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right) \][/tex]
For [tex]\(n = 3\)[/tex]:
[tex]\[ S_3 = a_1 + a_2 + a_3 = -12 \][/tex]
Using the general term formula, we express [tex]\(a_2\)[/tex] and [tex]\(a_3\)[/tex]:
[tex]\[ a_2 = a_1 + d, \quad a_3 = a_1 + 2d \][/tex]
Substituting these into the sum:
[tex]\[ a_1 + (a_1 + d) + (a_1 + 2d) = -12 \][/tex]
Simplifying:
[tex]\[ 3a_1 + 3d = -12 \][/tex]
Dividing by 3:
[tex]\[ a_1 + d = -4 \][/tex]
6. Solving the System of Equations:
We have two equations now:
[tex]\[ a_1 = -2d \][/tex]
[tex]\[ a_1 + d = -4 \][/tex]
Substitute [tex]\(a_1 = -2d\)[/tex] into [tex]\(a_1 + d = -4\)[/tex]:
[tex]\[ -2d + d = -4 \][/tex]
[tex]\[ -d = -4 \][/tex]
[tex]\[ d = 4 \][/tex]
Using [tex]\(d = 4\)[/tex] to find [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = -2d = -2(4) = -8 \][/tex]
7. Form of the Arithmetic Sequence (AP):
The general term [tex]\(a_n\)[/tex] for the arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1) d \][/tex]
Substituting the values we found:
[tex]\[ a_n = -8 + (n-1) \cdot 4 \][/tex]
Simplifying:
[tex]\[ a_n = -8 + 4n - 4 \][/tex]
[tex]\[ a_n = 4n - 12 \][/tex]
Thus, the arithmetic sequence (AP) is:
[tex]\[ a_n = 4n - 12 \][/tex]
1. Using the General Formula of an Arithmetic Sequence:
For an arithmetic sequence, the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) can be expressed as:
[tex]\[ a_n = a_1 + (n-1) d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
2. Getting Information from the Problem Statements:
- The sixth term ([tex]\(a_6\)[/tex]) is 3 times the fourth term ([tex]\(a_4\)[/tex]):
[tex]\[ a_6 = 3 \cdot a_4 \][/tex]
- The sum of the first three terms ([tex]\(a_1 + a_2 + a_3\)[/tex]) is -12:
[tex]\[ a_1 + a_2 + a_3 = -12 \][/tex]
3. Expressing [tex]\(a_6\)[/tex] and [tex]\(a_4\)[/tex] in Terms of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
Using the formula [tex]\(a_n = a_1 + (n-1) d\)[/tex]:
- The fourth term:
[tex]\[ a_4 = a_1 + 3d \][/tex]
- The sixth term:
[tex]\[ a_6 = a_1 + 5d \][/tex]
Substituting these into the given relation [tex]\(a_6 = 3 \cdot a_4\)[/tex]:
[tex]\[ a_1 + 5d = 3(a_1 + 3d) \][/tex]
4. Solving for [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
Expanding the equation we get:
[tex]\[ a_1 + 5d = 3a_1 + 9d \][/tex]
Rearranging to find [tex]\(a_1\)[/tex]:
[tex]\[ 5d - 9d = 3a_1 - a_1 \][/tex]
[tex]\[ -4d = 2a_1 \][/tex]
[tex]\[ a_1 = -2d \][/tex]
5. Using Sum of the First Three Terms:
The formula for the sum of the first [tex]\(n\)[/tex] terms [tex]\(S_n\)[/tex] of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right) \][/tex]
For [tex]\(n = 3\)[/tex]:
[tex]\[ S_3 = a_1 + a_2 + a_3 = -12 \][/tex]
Using the general term formula, we express [tex]\(a_2\)[/tex] and [tex]\(a_3\)[/tex]:
[tex]\[ a_2 = a_1 + d, \quad a_3 = a_1 + 2d \][/tex]
Substituting these into the sum:
[tex]\[ a_1 + (a_1 + d) + (a_1 + 2d) = -12 \][/tex]
Simplifying:
[tex]\[ 3a_1 + 3d = -12 \][/tex]
Dividing by 3:
[tex]\[ a_1 + d = -4 \][/tex]
6. Solving the System of Equations:
We have two equations now:
[tex]\[ a_1 = -2d \][/tex]
[tex]\[ a_1 + d = -4 \][/tex]
Substitute [tex]\(a_1 = -2d\)[/tex] into [tex]\(a_1 + d = -4\)[/tex]:
[tex]\[ -2d + d = -4 \][/tex]
[tex]\[ -d = -4 \][/tex]
[tex]\[ d = 4 \][/tex]
Using [tex]\(d = 4\)[/tex] to find [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = -2d = -2(4) = -8 \][/tex]
7. Form of the Arithmetic Sequence (AP):
The general term [tex]\(a_n\)[/tex] for the arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1) d \][/tex]
Substituting the values we found:
[tex]\[ a_n = -8 + (n-1) \cdot 4 \][/tex]
Simplifying:
[tex]\[ a_n = -8 + 4n - 4 \][/tex]
[tex]\[ a_n = 4n - 12 \][/tex]
Thus, the arithmetic sequence (AP) is:
[tex]\[ a_n = 4n - 12 \][/tex]
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