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 express this as the sum of all cubes minus twice the sum of the even cubes.
Step 2: Key Formula or Approach
1. $\sum n^3 = \left[\frac{n(n+1)}{2}\right]^2$.
2. $S = (1^3 + 2^3 + \dots + 15^3) - 2(2^3 + 4^3 + \dots + 14^3)$.
Step 3: Detailed Explanation
Sum of first 15 cubes: $\left[\frac{15 \times 16}{2}\right]^2 = (120)^2 = 14400$.
Sum of even cubes up to 14: $2^3(1^3 + 2^3 + \dots + 7^3) = 8 \times \left[\frac{7 \times 8}{2}\right]^2 = 8 \times (28)^2 = 8 \times 784 = 6272$.
Alternating sum: $14400 - 2(6272) = 14400 - 12544 = 1856$.
Step 4: Final Answer
The value is 1856.