Question

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

• A. 1856
• B. 1854
• C. 1786
• D. 1826

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the sum of an alternating series of cubes, we can rearrange the series into two parts: the sum of all cubes up to the last term and the subtraction of twice the sum of even cubes. Alternatively, a specific formula for alternating sums of cubes can be derived.
Step 2: Key Formula or Approach:
1. Sum of first \( n \) cubes: \( \sum_{r=1}^{n} r^3 = \left[ \frac{n(n+1)}{2} \right]^2 \)
2. Alternating sum \( S = \sum_{r=1}^{15} (-1)^{r-1} r^3 = (1^3 + 2^3 + \dots + 15^3) - 2(2^3 + 4^3 + \dots + 14^3) \)
Step 3: Detailed Explanation:
1. Calculate the sum of all cubes from 1 to 15: \[ \sum_{r=1}^{15} r^3 = \left[ \frac{15 \times 16}{2} \right]^2 = (120)^2 = 14400 \] 2. Calculate the sum of even cubes from 2 to 14: \[ 2^3 + 4^3 + \dots + 14^3 = 2^3(1^3 + 2^3 + \dots + 7^3) = 8 \left[ \frac{7 \times 8}{2} \right]^2 \] \[ = 8 \times (28)^2 = 8 \times 784 = 6272 \] 3. Find the alternating sum \( S \): \[ S = 14400 - 2(6272) \] \[ S = 14400 - 12544 \] \[ S = 1856 \]
Step 4: Final Answer:
The value of the series is 1856.