Answer :
Let's solve the problem step-by-step:
Given:
- The third term of an arithmetic progression (AP) is [tex]\(-40\)[/tex].
- The thirteenth term of the AP is [tex]\(5\)[/tex].
We need to find:
(a) The first term and common difference.
(b) The [tex]\(30^{\text{th}}\)[/tex] term.
(c) Which term of the AP is [tex]\(35\)[/tex].
### Step (a): Find the first term and common difference.
The formula for the [tex]\(n^{\text{th}}\)[/tex] term of an AP is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
1. For the third term ([tex]\(a_3 = -40\)[/tex]):
[tex]\[
a + 2d = -40
\][/tex]
2. For the thirteenth term ([tex]\(a_{13} = 5\)[/tex]):
[tex]\[
a + 12d = 5
\][/tex]
Now, we'll solve these two equations simultaneously.
- Equation 1: [tex]\( a + 2d = -40 \)[/tex]
- Equation 2: [tex]\( a + 12d = 5 \)[/tex]
Subtract Equation 1 from Equation 2:
[tex]\[ (a + 12d) - (a + 2d) = 5 - (-40) \][/tex]
This simplifies to:
[tex]\[ 10d = 45 \][/tex]
[tex]\[ d = 4.5 \][/tex]
Now that we have [tex]\(d\)[/tex], substitute it back into one of the equations to find [tex]\(a\)[/tex]. Use Equation 1:
[tex]\[ a + 2(4.5) = -40 \][/tex]
[tex]\[ a + 9 = -40 \][/tex]
[tex]\[ a = -49 \][/tex]
So, the first term [tex]\(a = -49\)[/tex] and the common difference [tex]\(d = 4.5\)[/tex].
### Step (b): Find the [tex]\(30^{\text{th}}\)[/tex] term.
Using the formula for the [tex]\(n^{\text{th}}\)[/tex] term:
[tex]\[ a_{30} = a + 29d \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ a_{30} = -49 + 29(4.5) \][/tex]
[tex]\[ a_{30} = -49 + 130.5 \][/tex]
[tex]\[ a_{30} = 81.5 \][/tex]
So, the [tex]\(30^{\text{th}}\)[/tex] term is [tex]\(81.5\)[/tex].
### Step (c): Determine which term is 35.
Set up the equation for the [tex]\(n^{\text{th}}\)[/tex] term where it equals 35:
[tex]\[ a + (n-1)d = 35 \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ -49 + (n-1) \times 4.5 = 35 \][/tex]
Solving for [tex]\(n\)[/tex]:
[tex]\[ (n-1) \times 4.5 = 35 + 49 \][/tex]
[tex]\[ (n-1) \times 4.5 = 84 \][/tex]
[tex]\[ n-1 = \frac{84}{4.5} \][/tex]
[tex]\[ n-1 = 18.67 \][/tex]
[tex]\[ n = 19.67 \][/tex]
Since [tex]\(n\)[/tex] must be an integer, there is no exact term equal to [tex]\(35\)[/tex] in this AP. Therefore, no term in this particular AP equals 35 exactly.
Summary:
- First term ([tex]\(a\)[/tex]): [tex]\(-49\)[/tex]
- Common difference ([tex]\(d\)[/tex]): [tex]\(4.5\)[/tex]
- [tex]\(30^{\text{th}}\)[/tex] term: [tex]\(81.5\)[/tex]
- No term equals 35 exactly in this AP.
Given:
- The third term of an arithmetic progression (AP) is [tex]\(-40\)[/tex].
- The thirteenth term of the AP is [tex]\(5\)[/tex].
We need to find:
(a) The first term and common difference.
(b) The [tex]\(30^{\text{th}}\)[/tex] term.
(c) Which term of the AP is [tex]\(35\)[/tex].
### Step (a): Find the first term and common difference.
The formula for the [tex]\(n^{\text{th}}\)[/tex] term of an AP is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
1. For the third term ([tex]\(a_3 = -40\)[/tex]):
[tex]\[
a + 2d = -40
\][/tex]
2. For the thirteenth term ([tex]\(a_{13} = 5\)[/tex]):
[tex]\[
a + 12d = 5
\][/tex]
Now, we'll solve these two equations simultaneously.
- Equation 1: [tex]\( a + 2d = -40 \)[/tex]
- Equation 2: [tex]\( a + 12d = 5 \)[/tex]
Subtract Equation 1 from Equation 2:
[tex]\[ (a + 12d) - (a + 2d) = 5 - (-40) \][/tex]
This simplifies to:
[tex]\[ 10d = 45 \][/tex]
[tex]\[ d = 4.5 \][/tex]
Now that we have [tex]\(d\)[/tex], substitute it back into one of the equations to find [tex]\(a\)[/tex]. Use Equation 1:
[tex]\[ a + 2(4.5) = -40 \][/tex]
[tex]\[ a + 9 = -40 \][/tex]
[tex]\[ a = -49 \][/tex]
So, the first term [tex]\(a = -49\)[/tex] and the common difference [tex]\(d = 4.5\)[/tex].
### Step (b): Find the [tex]\(30^{\text{th}}\)[/tex] term.
Using the formula for the [tex]\(n^{\text{th}}\)[/tex] term:
[tex]\[ a_{30} = a + 29d \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ a_{30} = -49 + 29(4.5) \][/tex]
[tex]\[ a_{30} = -49 + 130.5 \][/tex]
[tex]\[ a_{30} = 81.5 \][/tex]
So, the [tex]\(30^{\text{th}}\)[/tex] term is [tex]\(81.5\)[/tex].
### Step (c): Determine which term is 35.
Set up the equation for the [tex]\(n^{\text{th}}\)[/tex] term where it equals 35:
[tex]\[ a + (n-1)d = 35 \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ -49 + (n-1) \times 4.5 = 35 \][/tex]
Solving for [tex]\(n\)[/tex]:
[tex]\[ (n-1) \times 4.5 = 35 + 49 \][/tex]
[tex]\[ (n-1) \times 4.5 = 84 \][/tex]
[tex]\[ n-1 = \frac{84}{4.5} \][/tex]
[tex]\[ n-1 = 18.67 \][/tex]
[tex]\[ n = 19.67 \][/tex]
Since [tex]\(n\)[/tex] must be an integer, there is no exact term equal to [tex]\(35\)[/tex] in this AP. Therefore, no term in this particular AP equals 35 exactly.
Summary:
- First term ([tex]\(a\)[/tex]): [tex]\(-49\)[/tex]
- Common difference ([tex]\(d\)[/tex]): [tex]\(4.5\)[/tex]
- [tex]\(30^{\text{th}}\)[/tex] term: [tex]\(81.5\)[/tex]
- No term equals 35 exactly in this AP.
