Find the number of terms in the A.P. 22, 28, 34, ..., 616:

Options

A. 80
B. 78
C. 99
D. 100

Answer

To find the number of terms in the arithmetic progression (A.P.) given as 22, 28, 34, ..., 616, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a = 22 \).
- The second term \( T_2 = 28 \).
- The common difference \( d = T_2 - T_1 = 28 - 22 = 6 \).

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( T_n \) of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
We need to find \( n \) when \( T_n = 616 \).

### Step 3: Set up the equation
Substituting the known values into the formula:
\[ 616 = 22 + (n - 1) \cdot 6 \]

### Step 4: Simplify the equation
First, subtract 22 from both sides:
\[ 616 - 22 = (n - 1) \cdot 6 \]
\[ 594 = (n - 1) \cdot 6 \]

### Step 5: Divide both sides by 6
\[ n - 1 = \frac{594}{6} \]
Calculating the right side:
\[ n - 1 = 99 \]

### Step 6: Solve for n
Now, add 1 to both sides:
\[ n = 99 + 1 = 100 \]

### Conclusion
The number of terms in the A.P. is \( n = 100 \).