Is 402 a term of the sequence : 8, 13, 18, 23, ................. ?

Answer

To determine if 402 is a term of the arithmetic progression (AP) given by the sequence: 8, 13, 18, 23, ..., we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the sequence is 8. The common difference \( d \) can be calculated as follows:
\[ d = a_2 - a_1 = 13 - 8 = 5 \]

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
We need to check if 402 can be expressed in this form.

### Step 3: Set up the equation
We set \( a_n = 402 \):
\[ 402 = 8 + (n - 1) \cdot 5 \]

### Step 4: Solve for \( n \)
First, simplify the equation:
\[ 402 = 8 + 5(n - 1) \]
Subtract 8 from both sides:
\[ 402 - 8 = 5(n - 1) \]
\[ 394 = 5(n - 1) \]
Now, divide both sides by 5:
\[ n - 1 = \frac{394}{5} \]
\[ n - 1 = 78.8 \]
Add 1 to both sides:
\[ n = 78.8 + 1 = 79.8 \]

### Step 5: Determine if \( n \) is an integer
Since \( n = 79.8 \) is not an integer, it indicates that 402 cannot be a term of the sequence.

### Conclusion
Thus, 402 is not a term of the given arithmetic progression.

---