Determine the AP whose 3rd term is 5 and the 7th term is 9.

Step 1: Define the general term of an AP

The formula for the $n^{\text{th}}$ term of an Arithmetic Progression (AP) is given by $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. We will use this to set up equations based on the given information.

Step 2: Formulate equations from given terms

We are given that the 3rd term ($a_3$) is 5 and the 7th term ($a_7$) is 9. Substituting $n=3$ and $a_3=5$ into the general formula gives our first equation. Similarly, substituting $n=7$ and $a_7=9$ gives our second equation. These form a system of two linear equations with two variables, $a$ and $d$.

Step 3: Solve the system of equations for d

To find the common difference $d$, we subtract the first equation from the second. This eliminates $a$, allowing us to solve for $d$. Performing the subtraction gives $4d=4$, which simplifies to $d=1$.

Step 4: Solve for a using the value of d

Now that we have the value of $d$, we can substitute it back into one of the original equations to find the first term $a$. Using the equation $a + 2d = 5$ and knowing $d=1$, we find $a+2(1)=5$, which simplifies to $a=3$.

Step 5: Determine the AP

With the first term $a=3$ and the common difference $d=1$, we can now write out the arithmetic progression. Each subsequent term is found by adding the common difference to the previous term.

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