If `t_(n)=(3+4n)` of an `ap`, then the sum of the its 15 terms is:

Answer

To find the sum of the first 15 terms of the arithmetic progression (AP) given by \( t_n = 3 + 4n \), we can follow these steps:

### Step 1: Identify the first term and common difference
The general term of the AP is given by:
\[ t_n = 3 + 4n \]
To find the first term \( t_1 \):
\[ t_1 = 3 + 4(1) = 3 + 4 = 7 \]
So, the first term \( a = 7 \).

Next, we find the second term \( t_2 \):
\[ t_2 = 3 + 4(2) = 3 + 8 = 11 \]
The common difference \( d \) is calculated as:
\[ d = t_2 - t_1 = 11 - 7 = 4 \]

### Step 2: Use the formula for the sum of the first \( n \) terms
The formula for the sum of the first \( n \) terms \( S_n \) of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Here, we need to find the sum of the first 15 terms, so \( n = 15 \).

### Step 3: Substitute the values into the formula
Substituting \( n = 15 \), \( a = 7 \), and \( d = 4 \) into the formula:
\[ S_{15} = \frac{15}{2} \times (2 \times 7 + (15 - 1) \times 4) \]
Calculating inside the parentheses:
\[ 2 \times 7 = 14 \]
\[ (15 - 1) \times 4 = 14 \times 4 = 56 \]
Now, add these values:
\[ 14 + 56 = 70 \]
Now substitute back into the sum formula:
\[ S_{15} = \frac{15}{2} \times 70 \]

### Step 4: Calculate the final sum
Calculating:
\[ S_{15} = \frac{15 \times 70}{2} = \frac{1050}{2} = 525 \]

### Conclusion
Thus, the sum of the first 15 terms of the given AP is:
\[ \boxed{525} \]

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