Verified Answer :
- Express the sum as \(\sum_{r=15}^{30} (r^2 + 2r)\).
- Use the summation formulas for \(\sum r^2\) and \(\sum r\) to evaluate \(\sum_{r=1}^{30} f(r)\) and \(\sum_{r=1}^{14} f(r)\).
- Calculate \(\sum_{r=1}^{30} (r^2 + 2r) = 10385\) and \(\sum_{r=1}^{14} (r^2 + 2r) = 1225\).
- Find the difference \(10385 - 1225 = \boxed{9160}\).
Explanation
- Understanding the Problem and Approach
The objective is to evaluate the summation \(\sum_{r=15}^{30} r(r+2)\). We can write the expression as a sum of two parts: \(\sum_{r=15}^{30} (r^2 + 2r) = \sum_{r=15}^{30} r^2 + \sum_{r=15}^{30} 2r\). The summation runs from \(r=15\) to \(r=30\), so it involves 16 terms. To calculate this efficiently, we can use the identity \(\sum_{r=m}^n f(r) = \sum_{r=1}^n f(r) - \sum_{r=1}^{m-1} f(r)\), where \(f(r) = r^2 + 2r\), \(m=15\), and \(n=30\).
- Applying Summation Formulas
To evaluate the sum \(\sum_{r=1}^n (r^2 + 2r)\), we use the standard summation formulas: \(\sum_{k=1}^n k = \frac{n(n+1)}{2}\) and \(\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}\).
Thus, \(\sum_{r=1}^n (r^2 + 2r) = \frac{n(n+1)(2n+1)}{6} + 2 \cdot \frac{n(n+1)}{2} = \frac{n(n+1)(2n+1)}{6} + n(n+1)\).
Simplifying this: \(\frac{n(n+1)(2n+1+6)}{6} = \frac{n(n+1)(2n+7)}{6}\).
- Performing the Calculation
Now we calculate the sum up to \(n=30\):
\(\sum_{r=1}^{30} (r^2 + 2r) = \frac{30(30+1)(2(30)+7)}{6} = \frac{30 \cdot 31 \cdot 67}{6} = 5 \cdot 31 \cdot 67 = 155 \cdot 67 = 10385\).
Next, we calculate the sum up to \(n=14\) (since we need to subtract the first 14 terms):
\(\sum_{r=1}^{14} (r^2 + 2r) = \frac{14(14+1)(2(14)+7)}{6} = \frac{14 \cdot 15 \cdot 35}{6} = 7 \cdot 5 \cdot 35 = 35 \cdot 35 = 1225\).
- Conclusion
The value of the original sum \(\sum_{r=15}^{30} r(r+2)\) is the difference:
\(10385 - 1225 = 9160\).
Therefore, the total value of the summation is \(\boxed{9160}\).
Examples
Imagine you are managing inventory for a warehouse where you store items in numbered rows from 15 to 30. If each row \(r\) contains \(r(r+2)\) boxes, you need the total count to plan your storage space. This problem illustrates how to calculate such totals systematically, which is a foundational skill in logistics and supply chain management for resource allocation and planning.