Question

The sum $1+ \frac{1^{3} +2^{3}}{1+2} + \frac{1^{3}+2^{3}+3^{3}}{1+2+3} +.... + \frac{1^{3} +2^{3}+3^{3} +....+15^{3}}{1+2+3+...+15} - \frac{1}{2} \left(1+2+3+...+15\right)$

• A. 1240
• B. 1860
• C. 660
• D. 620

Answer 1

The Correct Option is D

To solve this problem, we need to evaluate the given expression:

1 + \frac{1^{3} + 2^{3}}{1 + 2} + \frac{1^{3} + 2^{3} + 3^{3}}{1 + 2 + 3} + \cdots + \frac{1^{3} + 2^{3} + 3^{3} + \cdots + 15^{3}}{1 + 2 + 3 + \cdots + 15} - \frac{1}{2} \left(1 + 2 + 3 + \cdots + 15\right)

We can break it down into a series that involves evaluating a sequence and some summation formulas. Let's solve it step by step:

1. The expression has two main components: the sum of fractional sequences and the subtraction term. First, let's understand the fractional part:

   \frac{1^{3} + 2^{3} + \cdots + n^{3}}{1 + 2 + \cdots + n} \text{ for } n = 1 \text{ to } 15

2. Using known formulas:

   Sum of first n natural numbers: \frac{n(n+1)}{2}

   Sum of the cubes of first n natural numbers: \left(\frac{n(n+1)}{2}\right)^2

   Therefore, the fraction simplifies to:

   \frac{\left(\frac{n(n+1)}{2}\right)^2}{\frac{n(n+1)}{2}} = \frac{n(n+1)}{2}

3. Now, evaluate the original expression from n = 1 to n = 15:

   You will compute: 1 + \sum_{n=2}^{15} \frac{n(n+1)}{2} - \frac{1}{2} \times 120

   Note: The sum of first 15 natural numbers: \frac{15 \cdot 16}{2} = 120

4. Calculate the summation:

   \sum_{n=2}^{15} \frac{n(n+1)}{2} = \sum_{n=2}^{15} \frac{n^2 + n}{2}

   Split into two sums: = \frac{1}{2}\left( \sum_{n=2}^{15} n^2 + \sum_{n=2}^{15} n \right)

5. Use known summation formulas:

   Sum of squares: \sum_{n=1}^{15} n^2 = \frac{15 \cdot 16 \cdot 31}{6}

   Subtract 1^2 as the summation starts from 2: = \frac{15 \cdot 16 \cdot 31}{6} - 1

   Sum of natural numbers already computed: 120 (from 2 to 15 is 120 - 1 = 119)

6. Compute the numerical terms to find:

   \frac{1}{2} \left( \frac{15 \cdot 16 \cdot 31}{6} - 1 + 119 \right) - 60

   Simplify to get:

   Ultimately, this calculation will result in the final value of 620.

Based on these calculations, the answer is:

620