Question

The value of \( 1^3 - 2^3 + 3^3 - 4^3 + \cdots - 14^3 + 15^3 \) is equal to:

• A. 1852
• B. 1856
• C. 1860
• D. 1864

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is an alternating sum of cubes. We can solve this by grouping the terms or by using the formula for the sum of cubes and subtracting twice the sum of the even cubes.

Step 2: Key Formula or Approach:
1. $S = \sum_{r=1}^{15} (-1)^{r-1} r^3$. 2. $S = (1^3 + 2^3 + \dots + 15^3) - 2(2^3 + 4^3 + \dots + 14^3)$.

Step 3: Detailed Explanation:
1. Sum of all cubes up to 15: \[ \sum_{r=1}^{15} r^3 = \left[ \frac{15(16)}{2} \right]^2 = (120)^2 = 14400 \] 2. Sum of even cubes: $2^3 + 4^3 + \dots + 14^3 = 2^3(1^3 + 2^3 + \dots + 7^3)$. \[ 8 \times \left[ \frac{7(8)}{2} \right]^2 = 8 \times (28)^2 = 8 \times 784 = 6272 \] 3. Alternating sum: \[ S = 14400 - 2(6272) = 14400 - 12544 = 1856 \]

Step 4: Final Answer:
The value is 1856.