Find the sum of the 10 terms of the following series `-11, -8, -5, -2…….`

Answer

To find the sum of the first 10 terms of the series \(-11, -8, -5, -2, \ldots\), we will follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the series is \(-11\).

To find the common difference \(d\), we subtract the first term from the second term:
\[ d = t_2 - t_1 = -8 - (-11) = -8 + 11 = 3 \]

### Step 2: Confirm that the series is an arithmetic progression (AP)
We will check the difference between the subsequent terms:
- \(t_3 - t_2 = -5 - (-8) = -5 + 8 = 3\)
- \(t_4 - t_3 = -2 - (-5) = -2 + 5 = 3\)

Since the common difference \(d\) is consistent, the series is indeed an arithmetic progression.

### Step 3: Use the formula for the sum of the first \(n\) terms of an AP
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]
Where:
- \(n\) is the number of terms (which is 10),
- \(a\) is the first term (\(-11\)),
- \(d\) is the common difference (3).

### Step 4: Substitute the values into the formula
Substituting \(n = 10\), \(a = -11\), and \(d = 3\) into the formula:
\[ S_{10} = \frac{10}{2} \left(2 \times -11 + (10 - 1) \times 3\right) \]
\[ = 5 \left(-22 + 9 \times 3\right) \]
\[ = 5 \left(-22 + 27\right) \]
\[ = 5 \times 5 \]
\[ = 25 \]

### Final Answer
The sum of the first 10 terms of the series is \(25\).

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