Problem 3 - Calculus Volume 1

Question

In the following exercises, use the rules for sums of powers of integers to compute the sums. $$\sum_{i=5}^{10} i^{2}$$

Step-by-Step Solution

Answer

The sum \( \sum_{i=5}^{10} i^2 \) equals 355.

Step 1: Understand the Sum of Powers Formula

To solve the sum \( \sum_{i=5}^{10} i^2 \), you need to understand that it represents the sum of the squares of integers from 5 to 10. This can be rewritten in expanded form as \( 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 \).

Step 2: Calculate Each Term

Calculate each individual square: \( 5^2 = 25 \), \( 6^2 = 36 \), \( 7^2 = 49 \), \( 8^2 = 64 \), \( 9^2 = 81 \), and \( 10^2 = 100 \).

Step 3: Add the Squares Together

Add the results from Step 2 to find the sum: \( 25 + 36 + 49 + 64 + 81 + 100 = 355 \).

Key Concepts

Squares of Integers

Understanding the concept of squares of integers is crucial when working with sums of powers of integers. When we talk about the "square" of a number, we are referring to the number multiplied by itself. This is denoted mathematically as \( i^2 \) for a given integer \( i \). For example, the square of 5, represented as \( 5^2 \), is calculated as \( 5 \times 5 = 25 \).
Squares play a prominent role in various fields such as geometry, algebra, and even in real-life calculations like area measurements. Recognizing patterns in square numbers can greatly assist in simplifying numerical problems by reducing repetitive calculations.

Calculation of Sums

Once you understand how to find the square of an integer, the next step is calculating the sums of these squares. This involves a straightforward process: calculate each square individually, then add them together.

• Identify the range of integers you are evaluating. For example, from 5 to 10 in our exercise.
• Calculate the square for each integer in the sequence:
  \[ 5^2 = 25, \; 6^2 = 36, \; 7^2 = 49, \; 8^2 = 64, \; 9^2 = 81, \; 10^2 = 100 \]
• Add these squared values together to get the total sum:
  \[ 25 + 36 + 49 + 64 + 81 + 100 = 355 \]

Proper calculation requires attention to detail and can be double-checked for accuracy by understanding each step fully.

Sum of Squares Formula

For larger ranges or more complex calculations, using a general formula to find the sum of squares can save time and reduce errors. The sum of squares formula for the first \( n \) integers is given by: \[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \] Although the exercise focuses on a specific range, understanding this formula highlights its usefulness. It quickly computes sums without needing manual computation of each term, especially beneficial if dealing with a large sequence of numbers.
For textbook exercises, if a specific formula is not provided for a narrower range, calculating manually as shown earlier is effective. However, knowing the existence of such formulas is valuable for complex mathematical problems.