Find the value of the expression 1 – 6 + 2 – 7 + 3 – 8 + ....... to 100 terms

Options

A. `-250`
B. `-500`
C. `-450`
D. `-300`

Answer

To find the value of the expression \(1 - 6 + 2 - 7 + 3 - 8 + \ldots\) up to 100 terms, we can break down the sequence into two parts: the positive terms and the negative terms.

### Step 1: Identify the Positive and Negative Terms
The sequence alternates between positive and negative terms. The positive terms are:
- \(1, 2, 3, \ldots, 50\) (the first 50 terms)

The negative terms are:
- \(6, 7, 8, \ldots, 55\) (the next 50 terms)

### Step 2: Calculate the Sum of Positive Terms
The sum of the first \(n\) natural numbers is given by the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
For the first 50 positive terms:
\[ S_{50} = \frac{50 \times (50 + 1)}{2} = \frac{50 \times 51}{2} = 1275 \]

### Step 3: Calculate the Sum of Negative Terms
The negative terms can be expressed as:
- \(6, 7, 8, \ldots, 55\) which is an arithmetic series where:
- First term \(a = 6\)
- Last term \(l = 55\)
- Number of terms \(n = 50\)

The sum of an arithmetic series is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the values:
\[ S_{50} = \frac{50}{2} \times (6 + 55) = 25 \times 61 = 1525 \]

### Step 4: Combine the Sums
Now, we combine the sums of the positive and negative terms:
\[ \text{Total Sum} = \text{Sum of Positive Terms} - \text{Sum of Negative Terms} \]
\[ \text{Total Sum} = 1275 - 1525 = -250 \]

### Final Answer
The value of the expression \(1 - 6 + 2 - 7 + 3 - 8 + \ldots\) up to 100 terms is \(-250\).

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