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].
