To determine the arithmetic progression (AP) where the 3rd term is 5 and the 7th term is 9, follow these steps:
1. Understand the general formula for the nth term of an AP:
[tex]\[
T_n = a + (n-1) \cdot d
\][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
2. Write the equations for the given terms:
For the 3rd term ([tex]\(T_3\)[/tex]):
[tex]\[
T_3 = a + 2d = 5 \quad \text{(Equation 1)}
\][/tex]
For the 7th term ([tex]\(T_7\)[/tex]):
[tex]\[
T_7 = a + 6d = 9 \quad \text{(Equation 2)}
\][/tex]
3. Solve the system of linear equations formed by Equation 1 and Equation 2:
First, subtract Equation 1 from Equation 2 to eliminate [tex]\(a\)[/tex]:
[tex]\[
(a + 6d) - (a + 2d) = 9 - 5
\][/tex]
Simplify the equation:
[tex]\[
4d = 4 \implies d = 1
\][/tex]
4. Substitute the value of [tex]\(d\)[/tex] back into Equation 1 to find [tex]\(a\)[/tex]:
Using Equation 1:
[tex]\[
a + 2(1) = 5
\][/tex]
Simplify to find [tex]\(a\)[/tex]:
[tex]\[
a + 2 = 5 \implies a = 3
\][/tex]
5. Determine the arithmetic progression:
With [tex]\(a = 3\)[/tex] and [tex]\(d = 1\)[/tex], the AP can be written as:
[tex]\[
a, a+d, a+2d, \ldots
\][/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[
3, 3+1, 3+2 \cdot 1, \ldots \implies 3, 4, 5, 6, 7, 8, 9, \ldots
\][/tex]
Therefore, the arithmetic progression is:
[tex]\[
3, 4, 5, 6, 7, 8, 9, \ldots
\][/tex]
