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Sagot :
Of course! Let's solve this problem step-by-step based on the information provided about the arithmetic progression (A.P.).
### Given:
1. The 11th term [tex]\(a_{11}\)[/tex] is -31.
2. The 21st term [tex]\(a_{21}\)[/tex] is -71.
We will determine:
a) The first term ([tex]\(a\)[/tex]),
b) The common difference ([tex]\(d\)[/tex]),
c) The 15th term ([tex]\(a_{15}\)[/tex]).
### Step-by-Step Solution
An arithmetic progression (A.P.) is characterized by the formula for the [tex]\(n\)[/tex]-th term:
[tex]\[ a_n = a + (n-1)d \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term,
- [tex]\(a\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the position of the term.
#### Step 1: Setting up the equations
Using the given terms:
1. For the 11th term:
[tex]\[ a_{11} = a + 10d = -31 \][/tex]
2. For the 21st term:
[tex]\[ a_{21} = a + 20d = -71 \][/tex]
#### Step 2: Solve for the common difference ([tex]\(d\)[/tex])
We have two equations:
[tex]\[ 1) \quad a + 10d = -31 \][/tex]
[tex]\[ 2) \quad a + 20d = -71 \][/tex]
Subtracting equation (1) from equation (2) to eliminate [tex]\(a\)[/tex]:
[tex]\[ (a + 20d) - (a + 10d) = -71 - (-31) \][/tex]
[tex]\[ 20d - 10d = -71 + 31 \][/tex]
[tex]\[ 10d = -40 \][/tex]
Solving for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{-40}{10} \][/tex]
[tex]\[ d = -4.0 \][/tex]
So, the common difference [tex]\(d\)[/tex] is [tex]\(-4.0\)[/tex].
#### Step 3: Solve for the first term ([tex]\(a\)[/tex])
Substitute the value of [tex]\(d\)[/tex] back into equation (1):
[tex]\[ a + 10(-4.0) = -31 \][/tex]
[tex]\[ a - 40 = -31 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = -31 + 40 \][/tex]
[tex]\[ a = 9 \][/tex]
So, the first term [tex]\(a\)[/tex] is [tex]\(9\)[/tex].
#### Step 4: Find the 15th term ([tex]\(a_{15}\)[/tex])
Using the formula for the [tex]\(n\)[/tex]-th term:
[tex]\[ a_{15} = a + (15-1)d \][/tex]
[tex]\[ a_{15} = 9 + 14(-4.0) \][/tex]
[tex]\[ a_{15} = 9 - 56 \][/tex]
[tex]\[ a_{15} = -47.0 \][/tex]
So, the 15th term [tex]\(a_{15}\)[/tex] is [tex]\(-47.0\)[/tex].
### Summary of Results
a) The first term [tex]\(a\)[/tex] is [tex]\(9\)[/tex].
b) The common difference [tex]\(d\)[/tex] is [tex]\(-4.0\)[/tex].
c) The 15th term [tex]\(a_{15}\)[/tex] is [tex]\(-47.0\)[/tex].
### Given:
1. The 11th term [tex]\(a_{11}\)[/tex] is -31.
2. The 21st term [tex]\(a_{21}\)[/tex] is -71.
We will determine:
a) The first term ([tex]\(a\)[/tex]),
b) The common difference ([tex]\(d\)[/tex]),
c) The 15th term ([tex]\(a_{15}\)[/tex]).
### Step-by-Step Solution
An arithmetic progression (A.P.) is characterized by the formula for the [tex]\(n\)[/tex]-th term:
[tex]\[ a_n = a + (n-1)d \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term,
- [tex]\(a\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the position of the term.
#### Step 1: Setting up the equations
Using the given terms:
1. For the 11th term:
[tex]\[ a_{11} = a + 10d = -31 \][/tex]
2. For the 21st term:
[tex]\[ a_{21} = a + 20d = -71 \][/tex]
#### Step 2: Solve for the common difference ([tex]\(d\)[/tex])
We have two equations:
[tex]\[ 1) \quad a + 10d = -31 \][/tex]
[tex]\[ 2) \quad a + 20d = -71 \][/tex]
Subtracting equation (1) from equation (2) to eliminate [tex]\(a\)[/tex]:
[tex]\[ (a + 20d) - (a + 10d) = -71 - (-31) \][/tex]
[tex]\[ 20d - 10d = -71 + 31 \][/tex]
[tex]\[ 10d = -40 \][/tex]
Solving for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{-40}{10} \][/tex]
[tex]\[ d = -4.0 \][/tex]
So, the common difference [tex]\(d\)[/tex] is [tex]\(-4.0\)[/tex].
#### Step 3: Solve for the first term ([tex]\(a\)[/tex])
Substitute the value of [tex]\(d\)[/tex] back into equation (1):
[tex]\[ a + 10(-4.0) = -31 \][/tex]
[tex]\[ a - 40 = -31 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = -31 + 40 \][/tex]
[tex]\[ a = 9 \][/tex]
So, the first term [tex]\(a\)[/tex] is [tex]\(9\)[/tex].
#### Step 4: Find the 15th term ([tex]\(a_{15}\)[/tex])
Using the formula for the [tex]\(n\)[/tex]-th term:
[tex]\[ a_{15} = a + (15-1)d \][/tex]
[tex]\[ a_{15} = 9 + 14(-4.0) \][/tex]
[tex]\[ a_{15} = 9 - 56 \][/tex]
[tex]\[ a_{15} = -47.0 \][/tex]
So, the 15th term [tex]\(a_{15}\)[/tex] is [tex]\(-47.0\)[/tex].
### Summary of Results
a) The first term [tex]\(a\)[/tex] is [tex]\(9\)[/tex].
b) The common difference [tex]\(d\)[/tex] is [tex]\(-4.0\)[/tex].
c) The 15th term [tex]\(a_{15}\)[/tex] is [tex]\(-47.0\)[/tex].
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