Problem 49 - Precalculus Mathematics for Calculus

Question

Use a graphing calculator to evaluate the sum. $$\sum_{k=1}^{10} k^{2}$$

Step-by-Step Solution

Answer

The sum is 385.

Step 1: Understand the Problem

We are asked to evaluate the sum \( \sum_{k=1}^{10} k^2 \), which means we need to find the sum of the squares of the first 10 positive integers.

Step 2: Define the Expression

The sum \( \sum_{k=1}^{10} k^2 \) can be broken down as follows: \( 1^2 + 2^2 + 3^2 + \, \dots \, + 10^2 \). Each term represents the square of an integer from 1 to 10.

Step 3: Calculate Each Square

Calculate the square of each number from 1 to 10: 1. \( 1^2 = 1 \) 2. \( 2^2 = 4 \) 3. \( 3^2 = 9 \) 4. \( 4^2 = 16 \) 5. \( 5^2 = 25 \) 6. \( 6^2 = 36 \) 7. \( 7^2 = 49 \) 8. \( 8^2 = 64 \) 9. \( 9^2 = 81 \) 10. \( 10^2 = 100 \).

Step 4: Sum the Squares

Add all the squared terms together: - \( 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 \).

Step 5: Calculate the Total

The sum of the squares is: \( 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385 \).

Key Concepts

Graphing Calculator Use

Using a graphing calculator can make tasks like evaluating sums much simpler and more efficient. A graphing calculator is a powerful tool that not only allows you to perform basic arithmetic calculations but also helps in visualizing functions and solving more complex mathematical problems. For this particular exercise, the \( \sum_{k=1}^{10} k^2 \) sum of squares can be quickly calculated using the built-in summation functions available in most graphing calculators.

Here’s a straightforward way to do it:

• First, turn on your graphing calculator and ensure it's in the correct mode for doing arithmetic operations.
• You'll find a summation or \( \Sigma \) function in the math menu; navigate to it using the arrow keys.
• Set the lower limit (\