To address the problem, let's break it down into two parts, each dealing with one of the given arithmetic progressions (AP).
### First AP:
Given:
- The [tex]\(6^{\text{th}}\)[/tex] term of the AP is 19.
- The common difference, [tex]\(d\)[/tex], is 3.
#### Step 1: Determine the first term ([tex]\(a\)[/tex]).
The formula for the [tex]\(n^{\text{th}}\)[/tex] term of an AP is:
[tex]\[ a_n = a + (n - 1)d \][/tex]
Given, [tex]\(a_6 = 19\)[/tex]:
[tex]\[ 19 = a + (6 - 1) \cdot 3 \][/tex]
[tex]\[ 19 = a + 15 \][/tex]
Subtract 15 from both sides to solve for [tex]\(a\)[/tex]:
[tex]\[ a = 19 - 15 = 4 \][/tex]
Therefore, the first term [tex]\(a\)[/tex] of this AP is 4.
#### Step 2: Find the [tex]\(7^{\text{th}}\)[/tex] term ([tex]\(a_7\)[/tex]).
Using the formula:
[tex]\[ a_7 = a + (7 - 1)d \][/tex]
[tex]\[ a_7 = 4 + 6 \cdot 3 \][/tex]
[tex]\[ a_7 = 4 + 18 \][/tex]
[tex]\[ a_7 = 22 \][/tex]
The [tex]\(7^{\text{th}}\)[/tex] term of the first AP is 22.
### Second AP:
Given:
- The [tex]\(8^{\text{th}}\)[/tex] term of the AP is 44.
- The common difference, [tex]\(d\)[/tex], is 4.
#### Step 1: Determine the first term ([tex]\(a\)[/tex]).
Given, [tex]\(a_8 = 44\)[/tex]:
[tex]\[ 44 = a + (8 - 1) \cdot 4 \][/tex]
[tex]\[ 44 = a + 28 \][/tex]
Subtract 28 from both sides to solve for [tex]\(a\)[/tex]:
[tex]\[ a = 44 - 28 = 16 \][/tex]
Therefore, the first term [tex]\(a\)[/tex] of this AP is 16.
#### Step 2: Find the [tex]\(7^{\text{th}}\)[/tex] term ([tex]\(a_7\)[/tex]).
Using the formula:
[tex]\[ a_7 = a + (7 - 1)d \][/tex]
[tex]\[ a_7 = 16 + 6 \cdot 4 \][/tex]
[tex]\[ a_7 = 16 + 24 \][/tex]
[tex]\[ a_7 = 40 \][/tex]
The [tex]\(7^{\text{th}}\)[/tex] term of the second AP is 40.
### Summary:
- The [tex]\(7^{\text{th}}\)[/tex] term of the first AP is [tex]\(22\)[/tex].
- The [tex]\(7^{\text{th}}\)[/tex] term of the second AP is [tex]\(40\)[/tex].
