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

Answer

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

### Step 1: Identify the first term and common difference
The first term \(a\) is \(-5\). The second term is \(-8\) and the third term is \(-11\).

To find the common difference \(d\):
\[ d = \text{second term} - \text{first term} = -8 - (-5) = -3 \]

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

### Step 3: Use the formula for the nth term of an arithmetic progression
The nth term of an arithmetic progression can be expressed as:
\[ l = a + (n-1) \cdot d \]
Substituting the known values:
\[ -230 = -5 + (n-1)(-3) \]

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

### Step 5: Calculate the sum of the first \(n\) terms
The sum \(S_n\) of the first \(n\) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values:
\[ S_{76} = \frac{76}{2} \cdot (-5 + (-230)) \]
\[ = 38 \cdot (-235) \]

### Step 6: Perform the multiplication
Calculating:
\[ S_{76} = 38 \cdot (-235) = -8930 \]

### Final Answer
The sum of the series \(-5 + (-8) + (-11) + \ldots + (-230)\) is \(-8930\).

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