Is 313, a term of the A.P. 5, 10, 15, ..... ?

Answer

To determine if 313 is a term of the arithmetic progression (A.P.) 5, 10, 15, ..., we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term of the A.P. is:
- \( a = 5 \)

The common difference (d) can be calculated as:
- \( d = 10 - 5 = 5 \)

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term (a_n) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
We need to check if 313 can be expressed in this form.

### Step 3: Set up the equation
We set \( a_n = 313 \):
\[ 313 = 5 + (n - 1) \cdot 5 \]

### Step 4: Simplify the equation
Subtract 5 from both sides:
\[ 313 - 5 = (n - 1) \cdot 5 \]
\[ 308 = (n - 1) \cdot 5 \]

### Step 5: Solve for (n - 1)
Now, divide both sides by 5:
\[ n - 1 = \frac{308}{5} \]
\[ n - 1 = 61.6 \]

### Step 6: Solve for n
Add 1 to both sides to find n:
\[ n = 61.6 + 1 \]
\[ n = 62.6 \]

### Step 7: Determine if n is a positive integer
Since n must be a positive integer, and 62.6 is not an integer, we conclude that:
- 313 is **not** a term of the A.P. 5, 10, 15, ...

### Final Answer
Therefore, 313 is not a term of the arithmetic progression.

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