`1^3-2^3+3^3-4^3+........+9^3` is equal to

Options

A. 425
B. `-425`
C. 475
D. `-475`

Answer

To solve the expression \( S = 1^3 - 2^3 + 3^3 - 4^3 + 5^3 - 6^3 + 7^3 - 8^3 + 9^3 \), we can follow these steps:

### Step 1: Group the terms
We can group the terms in pairs, where each pair consists of one positive and one negative cube:
\[ S = (1^3 - 2^3) + (3^3 - 4^3) + (5^3 - 6^3) + (7^3 - 8^3) + 9^3 \]

### Step 2: Calculate each pair
Now, we calculate each of the pairs:
1. \( 1^3 - 2^3 = 1 - 8 = -7 \)
2. \( 3^3 - 4^3 = 27 - 64 = -37 \)
3. \( 5^3 - 6^3 = 125 - 216 = -91 \)
4. \( 7^3 - 8^3 = 343 - 512 = -169 \)

Now we can rewrite \( S \) with these results:
\[ S = -7 - 37 - 91 - 169 + 9^3 \]

### Step 3: Calculate \( 9^3 \)
Next, we calculate \( 9^3 \):
\[ 9^3 = 729 \]

### Step 4: Combine all terms
Now we can combine all the terms:
\[ S = -7 - 37 - 91 - 169 + 729 \]

### Step 5: Calculate the total
First, we sum the negative terms:
\[ -7 - 37 = -44 \]
\[ -44 - 91 = -135 \]
\[ -135 - 169 = -304 \]

Now add \( 729 \):
\[ S = -304 + 729 = 425 \]

### Final Answer
Thus, the value of \( S \) is:
\[ \boxed{425} \]

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