If the third term of an A.P. is 5 and the seventh terms is 9, find the 17th term.

Answer

To solve the problem step by step, we will use the formula for the nth term of an Arithmetic Progression (A.P.), which is given by:

\[ A_n = A + (n - 1)D \]

where:
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the term number.

### Step 1: Set up the equations for the given terms
We know:
- The third term \( A_3 = 5 \)
- The seventh term \( A_7 = 9 \)

Using the formula for the nth term, we can write:
1. For the third term:
\[ A_3 = A + (3 - 1)D = A + 2D = 5 \] (Equation 1)

2. For the seventh term:
\[ A_7 = A + (7 - 1)D = A + 6D = 9 \] (Equation 2)

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

We can solve these equations by elimination. 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 3: Substitute \( D \) back to find \( A \)
Now that we have \( D \), we can substitute it back into Equation 1 to find \( A \):
\[ A + 2(1) = 5 \]
This simplifies to:
\[ A + 2 = 5 \]
Subtracting 2 from both sides gives:
\[ A = 3 \]

### Step 4: Find the 17th term
Now, we can find the 17th term \( A_{17} \):
\[ A_{17} = A + (17 - 1)D = A + 16D \]
Substituting the values of \( A \) and \( D \):
\[ A_{17} = 3 + 16(1) = 3 + 16 = 19 \]

### Final Answer
The 17th term of the A.P. is \( 19 \).

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