Problem 49 - Algebra and Trigonometry

Question

\(49-54\) . 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 Sigma Notation

The expression \( \sum_{k=1}^{10} k^2 \) represents the sum of the squares of the first 10 natural numbers. This means you need to calculate \( 1^2 + 2^2 + 3^2 + \ldots + 10^2 \).

Step 2: Calculate Individual Squares

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

Step 3: Sum the Squares

Add up all the squared values calculated in Step 2: \( 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 \).

Step 4: Evaluate the Total Sum

Performing the addition from Step 3 gives: \( 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385 \).

Step 5: Verify Using a Graphing Calculator

Use a graphing calculator to input \( \sum_{k=1}^{10} k^2 \) to verify the calculation. The calculator should confirm that the sum is indeed 385.

Key Concepts

Sigma Notation

Summation notation, or sigma notation, is a very concise and efficient way to denote the sum of a sequence of terms. The symbol for summation is the Greek capital letter sigma, \( \Sigma \). In mathematical terms, this notation typically appears as \( \sum_{k=1}^{n} a_k \), where \( k \) is the index of summation, \( 1 \) is the starting point, \( n \) is the endpoint, and \( a_k \) is the general form of the sequence.

For example, in the exercise given, \( \sum_{k=1}^{10} k^2 \) tells us to calculate the sum of the squares of the first 10 natural numbers. Here, \( k^2 \) is the term of the sequence that we need to sum. Each term is shaped by squaring the corresponding value of \( k \) from 1 to 10.

• Sigma notation handles large sums effortlessly by showing structure and boundaries clearly.
• It's a powerful tool for expressing complex series in mathematical expressions.

Squares of Natural Numbers

The concept of squaring natural numbers plays an essential role in many mathematical calculations and concepts. Squaring a number means multiplying the number by itself. For instance, \( 3^2 = 3 \times 3 = 9 \). In our given exercise, each term to be summed is a square of a natural number from 1 to 10.

Here are the squares of the first ten natural numbers:

• 1 squared is 1: \( 1^2 = 1 \)
• 2 squared is 4: \( 2^2 = 4 \)
• 3 squared is 9: \( 3^2 = 9 \)
• ... and so on up to 10 squared is 100: \( 10^2 = 100 \).

Squaring numbers not only results in larger numbers, but also has interesting properties in algebra and geometry.

For example, squares are always positive or zero because any number multiplied by itself cannot be negative. This property ensures stability in mathematical outcomes.

Graphing Calculator

A graphing calculator is a handy tool for mathematical computations. These powerful devices can solve complex mathematical equations, perform statistical calculations, and graph functions with ease. In our exercise, after calculating the sum manually, a graphing calculator was used to verify the result.

With this device, you can quickly input the summation notation: \( \sum_{k=1}^{10} k^2 \), and it will do the hard work of adding each squared term one by one:

• You start by entering the summation formula.
• The calculator computes each square and adds them up.
• Finally, it displays the result, confirming your manual calculations.

Using a graphing calculator efficiently requires knowing how to access and utilize its various functions. It’s a great help in cross-checking your work and saves time when dealing with larger or more complex sets of numbers.