Find the sum of n-terms of the series `2.5 + 5.8 + 8.11 +……..`

Answer

To find the sum of the first n terms of the series \(2.5 + 5.8 + 8.11 + \ldots\), we first need to identify the pattern in the series and derive a general formula for the nth term.

### Step 1: Identify the nth term of the series

The series can be rewritten as:
- First term: \(2\)
- Second term: \(5\)
- Third term: \(8\)
- Fourth term: \(11\)

We can observe that the first term is \(2\), the second term is \(5\), the third term is \(8\), and the fourth term is \(11\). The pattern shows that each term increases by \(3\).

Thus, we can express the nth term \(T_n\) as:
\[ T_n = 2 + (n - 1) \cdot 3 = 3n - 1 \]

### Step 2: Write the sum of the first n terms

The sum of the first n terms \(S_n\) can be expressed as:
\[ S_n = T_1 + T_2 + T_3 + \ldots + T_n = \sum_{k=1}^{n} T_k \]

Substituting \(T_k\) into the sum:
\[ S_n = \sum_{k=1}^{n} (3k - 1) \]

### Step 3: Simplify the summation

We can separate the summation:
\[ S_n = \sum_{k=1}^{n} (3k) - \sum_{k=1}^{n} 1 \]

The first summation can be calculated as:
\[ \sum_{k=1}^{n} 3k = 3 \sum_{k=1}^{n} k = 3 \cdot \frac{n(n + 1)}{2} = \frac{3n(n + 1)}{2} \]

The second summation is simply:
\[ \sum_{k=1}^{n} 1 = n \]

Putting it all together:
\[ S_n = \frac{3n(n + 1)}{2} - n \]

### Step 4: Combine the terms

To combine the terms, we can write:
\[ S_n = \frac{3n(n + 1)}{2} - \frac{2n}{2} = \frac{3n(n + 1) - 2n}{2} = \frac{3n^2 + 3n - 2n}{2} = \frac{3n^2 + n}{2} \]

### Final Result

Thus, the sum of the first n terms of the series is:
\[ S_n = \frac{3n^2 + n}{2} \]