Find the sum of the first 16 terms of a sequence whose nth term is given by `t_(n)=5n-3`, where n is a nautral number.

Answer

To find the sum of the first 16 terms of the sequence defined by the nth term \( t_n = 5n - 3 \), we will follow these steps:

### Step 1: Identify the first term and the common difference
The first term \( t_1 \) can be calculated by substituting \( n = 1 \) into the formula:
\[ t_1 = 5(1) - 3 = 5 - 3 = 2 \]

Next, we calculate the second term \( t_2 \):
\[ t_2 = 5(2) - 3 = 10 - 3 = 7 \]

Now, we calculate the third term \( t_3 \):
\[ t_3 = 5(3) - 3 = 15 - 3 = 12 \]

From these calculations, we can see that the first three terms of the sequence are \( 2, 7, 12 \).

The common difference \( d \) can be calculated as:
\[ d = t_2 - t_1 = 7 - 2 = 5 \]

### Step 2: Use the formula for the sum of the first n terms of an arithmetic progression (AP)
The formula for the sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first 16 terms, so \( n = 16 \), \( a = 2 \), and \( d = 5 \):
\[ S_{16} = \frac{16}{2} \times (2 \times 2 + (16 - 1) \times 5) \]

Calculating this step-by-step:
1. Calculate \( \frac{16}{2} = 8 \)
2. Calculate \( 2 \times 2 = 4 \)
3. Calculate \( 16 - 1 = 15 \)
4. Calculate \( 15 \times 5 = 75 \)
5. Now substitute these values back into the equation:
\[ S_{16} = 8 \times (4 + 75) \]
6. Calculate \( 4 + 75 = 79 \)
7. Finally, calculate \( 8 \times 79 = 632 \)

### Conclusion
Thus, the sum of the first 16 terms of the sequence is:
\[ \boxed{632} \]