Is 310 a term of the A.P 3,8,13,18,…………..?

Answer

To determine if 310 is a term of the arithmetic progression (A.P.) given by 3, 8, 13, 18, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a_1 \) of the A.P. is 3. The common difference \( d \) can be calculated as follows:
\[ d = 8 - 3 = 5 \]

### Step 2: Write the formula for the n-th term of an A.P.
The n-th term \( a_n \) of an A.P. can be expressed using the formula:
\[ a_n = a_1 + (n - 1) \cdot d \]
Substituting the values we have:
\[ a_n = 3 + (n - 1) \cdot 5 \]

### Step 3: Set the n-th term equal to 310
We need to check if 310 can be expressed as a term in this A.P. Therefore, we set:
\[ 3 + (n - 1) \cdot 5 = 310 \]

### Step 4: Solve for \( n \)
Rearranging the equation:
\[ (n - 1) \cdot 5 = 310 - 3 \]
\[ (n - 1) \cdot 5 = 307 \]
Now, divide both sides by 5:
\[ n - 1 = \frac{307}{5} \]
\[ n - 1 = 61.4 \]
Adding 1 to both sides gives:
\[ n = 61.4 + 1 = 62.4 \]

### Step 5: Determine if \( n \) is a whole number
Since \( n = 62.4 \) is not a whole number, it indicates that 310 cannot be a term of the A.P.

### Conclusion
Thus, we conclude that 310 is not a term of the given arithmetic progression.

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