For an A.P., `t_(3) = 12 `and `t_(7) = 24` . What is the value of `t_(5)` ?

Options

A. 18
B. 17
C. 16
D. 15

Answer

To find the value of \( t_5 \) in the given arithmetic progression (A.P.), we start with the information provided about the third and seventh terms.

### Step 1: Define the terms of the A.P.
In an arithmetic progression, the \( n \)-th term is given by the formula:
\[ t_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write equations for the given terms.
From the problem, we know:
- \( t_3 = 12 \)
- \( t_7 = 24 \)

Using the formula for the \( n \)-th term, we can write:
\[ t_3 = a + 2d = 12 \quad \text{(Equation 1)} \]
\[ t_7 = a + 6d = 24 \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 1 from Equation 2.
To eliminate \( a \) and find \( d \), we subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 2d) = 24 - 12 \]
This simplifies to:
\[ 4d = 12 \]
Now, solve for \( d \):
\[ d = \frac{12}{4} = 3 \]

### Step 4: Substitute \( d \) back into Equation 1 to find \( a \).
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 2(3) = 12 \]
This simplifies to:
\[ a + 6 = 12 \]
Now, solve for \( a \):
\[ a = 12 - 6 = 6 \]

### Step 5: Find \( t_5 \).
Now that we have both \( a \) and \( d \), we can find \( t_5 \):
\[ t_5 = a + 4d \]
Substituting the values of \( a \) and \( d \):
\[ t_5 = 6 + 4(3) = 6 + 12 = 18 \]

### Final Answer:
The value of \( t_5 \) is \( 18 \).

---