Find the volume of the following expression.
1 - 2 - 3 + 2 - 3 - 4 + ·········+ 100 terms

Options

A. `-626 `
B. `-622 `
C. `-624 `
D. `-628 `

Answer

To find the volume of the given expression \(1 - 2 - 3 + 2 - 3 - 4 + \ldots\) up to 100 terms, we can follow these steps:

### Step 1: Identify the pattern in the expression
The expression alternates between positive and negative terms. We can break it down into three parts based on the sequence of terms.

### Step 2: Count the number of positive and negative terms
From the given expression, we can see that there are 34 positive terms and 66 negative terms in the first 100 terms.

### Step 3: Break down the expression into three separate sums
1. **Positive terms**: \(1, 2, 3, \ldots, 34\)
2. **First set of negative terms**: \(-2, -3, -4, \ldots, -34\)
3. **Second set of negative terms**: \(-3, -4, -5, \ldots, -35\)

### Step 4: Calculate the sum of each part
1. **Sum of positive terms**:
\[ S_1 = 1 + 2 + 3 + \ldots + 34 \]
Using the formula for the sum of the first \(n\) natural numbers:
\[ S_n = \frac{n(n + 1)}{2} \]
For \(n = 34\):
\[ S_1 = \frac{34 \times 35}{2} = 595 \]

2. **Sum of the first set of negative terms**:
\[ S_2 = - (2 + 3 + 4 + \ldots + 34) \]
This can be calculated as:
\[ S_2 = - \left( \frac{34 \times 35}{2} - 1 \right) = - (595 - 1) = -594 \]

3. **Sum of the second set of negative terms**:
\[ S_3 = - (3 + 4 + 5 + \ldots + 35) \]
This can be calculated as:
\[ S_3 = - \left( \frac{35 \times 36}{2} - (1 + 2) \right) = - \left( 630 - 3 \right) = -627 \]

### Step 5: Combine the sums
Now, we can combine all three sums:
\[ \text{Total Sum} = S_1 + S_2 + S_3 = 595 - 594 - 627 \]
Calculating this gives:
\[ \text{Total Sum} = 595 - 594 - 627 = -626 \]

### Final Answer
Thus, the volume of the given expression is \(-626\).

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