−5 + (−8) + (−11) + ....... + (−230) is :

Q: −5 + (−8) + (−11) + ....... + (−230) is :

To solve the problem of finding the sum of the series $-5 + (-8) + (-11) + \ldots + (-230)$, we need to recognize that this is an arithmetic series. An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. ### Step-by-Step Explanation: 1. **Identify the First Term and Common Difference:** - The first term ($a$) of the series is $-5$. - The common difference ($d$) is the difference between consecutive terms, which is $-8 - (-5) = -3$. 2. **Find the Last Term:** - The last term ($l$) of the series is given as $-230$. 3. **Determine the Number of Terms ($n$):** - The formula for the $n$-th term of an arithmetic sequence is given by: \[ a_n = a + (n-1) \cdot d \] - Setting $a_n = -230$, we have: \[ -230 = -5 + (n-1) \cdot (-3) \] \[ -230 = -5 - 3(n-1) \] \[ -230 = -5 - 3n + 3 \] \[ -230 = -2 - 3n \] \[ -228 = -3n \] \[ n = 76 \] 4. **Calculate the Sum of the Series:** - The sum $S_n$ of an arithmetic series is given by: \[ S_n = \frac{n}{2} \cdot (a + l) \] - Substituting the known values: \[ S_{76} = \frac{76}{2} \cdot (-5 + (-230)) \] \[ S_{76} = 38 \cdot (-235) \] \[ S_{76} = 38 \cdot -235 = -8930 \] ### Conclusion: - **Option 1: -8930** is the correct answer because the calculated sum of the series is $-8930$. ### Explanation of Incorrect Options: - **Option 2: 8930** is incorrect because the sum of the series is negative, not positive. - **Option 3: 8989** is incorrect because it does not match the calculated sum of $-8930$. - **Option 4: 8900** is incorrect because it is not the exact sum of the series, which is $-8930$. Thus, the correct answer is indeed Option 1: $-8930$.

Question

Accepted Answer

To solve the problem of finding the sum of the series $-5 + (-8) + (-11) + \ldots + (-230)$, we need to recognize that this is an arithmetic series. An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant.

### Step-by-Step Explanation:

1. **Identify the First Term and Common Difference:**
   - The first term ($a$) of the series is $-5$.
   - The common difference ($d$) is the difference between consecutive terms, which is $-8 - (-5) = -3$.

2. **Find the Last Term:**
   - The last term ($l$) of the series is given as $-230$.

3. **Determine the Number of Terms ($n$):**
   - The formula for the $n$-th term of an arithmetic sequence is given by:
     $$
     a_n = a + (n-1) \cdot d
     $$
   - Setting $a_n = -230$, we have:
     $$
     -230 = -5 + (n-1) \cdot (-3)
     $$
     $$
     -230 = -5 - 3(n-1)
     $$
     $$
     -230 = -5 - 3n + 3
     $$
     $$
     -230 = -2 - 3n
     $$
     $$
     -228 = -3n
     $$
     $$
     n = 76
     $$

4. **Calculate the Sum of the Series:**
   - The sum $S_n$ of an arithmetic series is given by:
     $$
     S_n = \frac{n}{2} \cdot (a + l)
     $$
   - Substituting the known values:
     $$
     S_{76} = \frac{76}{2} \cdot (-5 + (-230))
     $$
     $$
     S_{76} = 38 \cdot (-235)
     $$
     $$
     S_{76} = 38 \cdot -235 = -8930
     $$

### Conclusion:

- **Option 1: -8930** is the correct answer because the calculated sum of the series is $-8930$.

### Explanation of Incorrect Options:

- **Option 2: 8930** is incorrect because the sum of the series is negative, not positive.
- **Option 3: 8989** is incorrect because it does not match the calculated sum of $-8930$.
- **Option 4: 8900** is incorrect because it is not the exact sum of the series, which is $-8930$.

Thus, the correct answer is indeed Option 1: $-8930$.