IF `A=1-10+3-12+5-14+7+…` upto 60 terms , then what is the value of A?

Options

A. `-360`
B. `-310`
C. `-240`
D. `-270`

Answer

To solve the problem \( A = 1 - 10 + 3 - 12 + 5 - 14 + 7 - 16 + \ldots \) up to 60 terms, we can break it down into two separate sequences: one for the positive terms and one for the negative terms.

### Step-by-step Solution:

1. **Identify the Positive and Negative Terms**:
- The positive terms are: \( 1, 3, 5, 7, \ldots \)
- The negative terms are: \( 10, 12, 14, 16, \ldots \)

2. **Count the Number of Terms**:
- Since the total number of terms is 60, there will be 30 positive terms and 30 negative terms.

3. **Identify the Sequences**:
- The positive terms form an arithmetic progression (AP) with:
- First term \( a = 1 \)
- Common difference \( d = 2 \)
- Number of terms \( n = 30 \)
- The negative terms also form an AP with:
- First term \( a = 10 \)
- Common difference \( d = 2 \)
- Number of terms \( n = 30 \)

4. **Sum of the Positive Terms**:
- The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
- For the positive terms:
\[ S_{positive} = \frac{30}{2} \times (2 \times 1 + (30 - 1) \times 2) = 15 \times (2 + 58) = 15 \times 60 = 900 \]

5. **Sum of the Negative Terms**:
- For the negative terms:
\[ S_{negative} = \frac{30}{2} \times (2 \times 10 + (30 - 1) \times 2) = 15 \times (20 + 58) = 15 \times 78 = 1170 \]

6. **Calculate the Value of A**:
- Now, we can find \( A \):
\[ A = S_{positive} - S_{negative} = 900 - 1170 = -270 \]

Thus, the value of \( A \) is \( -270 \).