Find the sums given below:
`-5 + (-8) + (-11) + ...+ (-230)`

Answer

To find the sum of the series \(-5 + (-8) + (-11) + ... + (-230)\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) is \(-5\). To find the common difference \(d\), we subtract the first term from the second term:
\[ d = -8 - (-5) = -8 + 5 = -3 \]

### Step 2: Identify the last term
The last term \(l\) is given as \(-230\).

### Step 3: Use the formula for the \(n\)-th term of an arithmetic progression
The formula for the \(n\)-th term \(T_n\) of an arithmetic progression is:
\[ T_n = a + (n-1) \cdot d \]
We need to find \(n\) such that \(T_n = -230\):
\[ -230 = -5 + (n-1)(-3) \]

### Step 4: Solve for \(n\)
Rearranging the equation:
\[ -230 + 5 = (n-1)(-3) \]
\[ -225 = (n-1)(-3) \]
Dividing both sides by \(-3\):
\[ n - 1 = 75 \]
Adding 1 to both sides:
\[ n = 76 \]

### Step 5: Use the formula for the sum of the first \(n\) terms
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{76} = \frac{76}{2} \cdot (-5 + (-230)) \]
\[ S_{76} = 38 \cdot (-235) \]

### Step 6: Calculate the sum
Now, calculating \(38 \cdot (-235)\):
\[ S_{76} = 38 \cdot (-235) = -8930 \]

### Final Answer
Thus, the sum of the series is:
\[ \boxed{-8930} \]