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

Answer

To determine the arithmetic progression whose 3rd term is 5 and 7th term is 9, we can follow these steps:

### Step 1: Understand the formula for the nth term of an arithmetic progression (AP)
The nth term of an arithmetic progression can be expressed using the formula:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 2: Set up equations for the given terms
We know:
- The 3rd term (\( a_3 \)) is 5.
- The 7th term (\( a_7 \)) is 9.

Using the formula for the 3rd term:
\[ a_3 = a + (3 - 1) \cdot d \]
\[ a + 2d = 5 \] (Equation 1)

Using the formula for the 7th term:
\[ a_7 = a + (7 - 1) \cdot d \]
\[ a + 6d = 9 \] (Equation 2)

### Step 3: Solve the equations simultaneously
Now we have two equations:
1. \( a + 2d = 5 \)
2. \( a + 6d = 9 \)

To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 2d) = 9 - 5 \]
This simplifies to:
\[ 4d = 4 \]
Dividing both sides by 4 gives:
\[ d = 1 \]

### Step 4: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 2(1) = 5 \]
\[ a + 2 = 5 \]
Subtracting 2 from both sides gives:
\[ a = 3 \]

### Step 5: Write the arithmetic progression
Now that we have both \( a \) and \( d \):
- First term \( a = 3 \)
- Common difference \( d = 1 \)

The arithmetic progression can be written as:
- \( a_1 = a = 3 \)
- \( a_2 = a + d = 3 + 1 = 4 \)
- \( a_3 = a + 2d = 3 + 2 = 5 \)
- \( a_4 = a + 3d = 3 + 3 = 6 \)
- \( a_5 = a + 4d = 3 + 4 = 7 \)
- \( a_6 = a + 5d = 3 + 5 = 8 \)
- \( a_7 = a + 6d = 3 + 6 = 9 \)

Thus, the arithmetic progression is:
\[ 3, 4, 5, 6, 7, 8, 9 \]