Find the number of terms in the AP 16,22,28,34,…..616

Options

A. 80
B. 78
C. 99
D. 101

Answer

To find the number of terms in the arithmetic progression (AP) given by 16, 22, 28, 34, ..., 616, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) is 16. The common difference \( d \) can be calculated as:
\[ d = \text{second term} - \text{first term} = 22 - 16 = 6 \]

### Step 2: Write the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( n \) is the number of terms,
- \( d \) is the common difference.

### Step 3: Set up the equation for the last term
In this case, we know that the last term \( a_n \) is 616. Therefore, we can set up the equation:
\[ 616 = 16 + (n - 1) \cdot 6 \]

### Step 4: Solve for \( n \)
First, we will isolate \( (n - 1) \cdot 6 \):
\[ 616 - 16 = (n - 1) \cdot 6 \]
\[ 600 = (n - 1) \cdot 6 \]
Now, divide both sides by 6:
\[ n - 1 = \frac{600}{6} = 100 \]
Now, add 1 to both sides to find \( n \):
\[ n = 100 + 1 = 101 \]

### Conclusion
The number of terms in the AP series is \( n = 101 \).

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