Problem 48 - Algebra and Trigonometry

Question

Use a graphing calculator to evaluate the sum. $$\sum_{k=1}^{100}(3 k+4)$$

Step-by-Step Solution

Answer

The sum \( \sum_{k=1}^{100}(3k+4) \) is 16,150.

Step 1: Understanding the Problem

The exercise requires us to find the sum of the expression \(3k+4\) as \(k\) ranges from \(1\) to \(100\). This expression is a linear function of \(k\). We need to evaluate the sum \(\sum_{k=1}^{100}(3k+4)\).

Step 2: Set Up the Summation on the Calculator

Access the summation function on your graphing calculator. This is often found under the 'Math' menu. Set up the summation with lower limit \(k=1\), upper limit \(k=100\), and the expression \(3k+4\).

Step 3: Execution of the Formula

Use the calculator to compute the sum. Once entered correctly, press 'Enter' to evaluate the sum. The calculator handles the calculation of each individual term and adds them together.

Step 4: Analysis of Result

After the graphing calculator processes the input, it will display the sum of the sequence. This represents the total when each term \(3k+4\) is summed from \(k=1\) to \(k=100\).

Key Concepts

Graphing Calculator

A graphing calculator is a versatile tool that helps with complex mathematical calculations, including summation. To evaluate a series like \( \sum_{k=1}^{100}(3k+4) \), you can use its summation function. Here’s how:

• Access the 'Math' menu.
• Navigate to the summation (\( \Sigma \)) function.
• Enter the parameters: the lower limit \( k = 1 \), the upper limit \( k = 100 \), and the function \( 3k + 4 \).

Once these are set, the graphing calculator swiftly computes the sum, eliminating the need for manual calculations. It can handle the processing of each term accurately and efficiently, providing the total sum for the sequence.

Linear Function

A linear function is a polynomial function of degree one, which can be expressed as \( f(x) = ax + b \). In our context, the expression \( 3k + 4 \) is a linear function of \( k \). Here's how it breaks down:

• The term \( 3k \) indicates the rate of change, also known as the slope.
• The constant \( 4 \) demonstrates the y-intercept, or where the function crosses the y-axis when \( k = 0 \).

Linear functions are straightforward because they have a constant rate of change. This means as \( k \) increases or decreases, \( 3k + 4 \) will change at a consistent rate, making these types of functions predictable and easy to graph or sum.

Series Evaluation

Series evaluation involves finding the sum of a sequence of terms. For the series \( \sum_{k=1}^{100}(3k+4) \), we use a structured approach:

• Identify the sequence's initial term by substituting \( k = 1 \) into \( 3k + 4 \), yielding 7.
• Recognize that each subsequent term increases by 3, as determined by the linear function \( 3k + 4 \).

Evaluating this sum means adding up all these calculated values as \( k \) spans from 1 to 100. A graphing calculator streamlines this process, utilizing its built-in functions to perform the calculations rapidly and accurately. This ensures precise results without manual addition or potential errors, making series evaluation accessible even for large sums.