Let's solve the problem step-by-step to find the [tex]$24^{\text{th}}$[/tex] term of the arithmetic progression (AP) where the [tex]$3^{\text{rd}}$[/tex] term is 32 and the [tex]$7^{\text{th}}$[/tex] term is 72.
1. Define the General Formula for an AP:
The general formula for the [tex]$n^{\text{th}}$[/tex] term of an arithmetic progression (AP) is given by:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
Where:
- [tex]\(a_n\)[/tex] is the [tex]$n^{\text{th}}$[/tex] term,
- [tex]\(a\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
2. Set Up the Equations:
Given:
[tex]\[
a + 2d = 32 \quad \text{(3rd term)} \quad ... (1)
\][/tex]
[tex]\[
a + 6d = 72 \quad \text{(7th term)} \quad ... (2)
\][/tex]
3. Solve the Equations:
To find the common difference [tex]\(d\)[/tex], subtract equation (1) from equation (2):
[tex]\[
(a + 6d) - (a + 2d) = 72 - 32
\][/tex]
[tex]\[
4d = 40
\][/tex]
[tex]\[
d = \frac{40}{4} = 10
\][/tex]
4. Find the First Term [tex]\(a\)[/tex]:
Substitute [tex]\(d = 10\)[/tex] back into equation (1):
[tex]\[
a + 2d = 32
\][/tex]
[tex]\[
a + 2 \cdot 10 = 32
\][/tex]
[tex]\[
a + 20 = 32
\][/tex]
[tex]\[
a = 32 - 20 = 12
\][/tex]
5. Find the [tex]$24^{\text{th}}$[/tex] Term:
Now, we use the general formula with [tex]\(a = 12\)[/tex] and [tex]\(d = 10\)[/tex]:
[tex]\[
a_{24} = a + (24-1) \cdot d
\][/tex]
[tex]\[
a_{24} = 12 + 23 \cdot 10
\][/tex]
[tex]\[
a_{24} = 12 + 230
\][/tex]
[tex]\[
a_{24} = 242
\][/tex]
So, the [tex]$24^{\text{th}}$[/tex] term of the AP is 242.
