Problem 56 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The sum is 15550.

Step 1: Understanding the Summation

The given expression is a summation, represented as \( \sum_{k=1}^{100}(3k+4) \). It means you need to calculate the sum of the expression \( 3k+4 \) for each integer value of \( k \) from 1 to 100.

Step 2: Identify the Formula to Use

Notice that the expression inside the summation has a form \( ax+b \). Here, \( a = 3 \) and \( b = 4 \). You can split the summation into separate sums: \[ \sum_{k=1}^{100}(3k+4) = \sum_{k=1}^{100}3k + \sum_{k=1}^{100}4 \] This allows us to evaluate each part individually.

Step 3: Calculate the Sum of Linear Term

First, calculate \( \sum_{k=1}^{100}3k \). The sum of the first \( n \) integers is \( \frac{n(n+1)}{2} \). Applying this for 3 times each integer:\[ \sum_{k=1}^{100}3k = 3 \times \frac{100(100+1)}{2} = 3 \times 5050 = 15150 \]

Step 4: Calculate the Sum of Constant Term

Next, calculate the sum of the constant term \( \sum_{k=1}^{100}4 \). Since 4 is constant and added 100 times:\[ \sum_{k=1}^{100}4 = 4 \times 100 = 400 \]

Step 5: Combine the Results

Now, add the results from Steps 3 and 4:\[ \sum_{k=1}^{100}(3k+4) = 15150 + 400 = 15550 \]

Key Concepts

graphing calculator

A graphing calculator is a powerful tool in mathematics, designed to plot graphs and solve equations with ease. When dealing with summation problems such as \( \sum_{k=1}^{100}(3k+4) \), a graphing calculator can simplify the process significantly. Instead of manually calculating each term, a graphing calculator can automate this process.

Graphing calculators have specific functions that allow users to input summation expressions. With the proper syntax, these calculators can compute the entire sum efficiently. To do this, you typically access the summation function in the calculator's menu, input your expression such as \( 3k+4 \), and specify the range from 1 to 100.

Using a graphing calculator to evaluate summations saves time and reduces errors. It is especially useful for complex or lengthy expressions, where hand calculations would be prone to mistakes.

linear expression

A linear expression is a mathematical expression involving variables raised to the power of one. These expressions have the general form \( ax + b \), where \( a \) and \( b \) are constants. In the context of the summation \( \sum_{k=1}^{100}(3k+4) \), the linear expression is \( 3k + 4 \).

Linear expressions like \( 3k + 4 \) are composed of a variable term, in this case represented by \( 3k \), and a constant term, \( 4 \).

• The coefficient \( 3 \) indicates how much \( k \) is scaled.
• The constant term \( 4 \) implies that each value of the linear term \( 3k \) is adjusted by adding 4.

Understanding linear expressions is vital because they form the basis for solving equations and evaluating summations in algebra.

sum of integers

Calculating the sum of integers is a common mathematical operation and involves adding a series of sequential whole numbers. For instance, when tackling \( \sum_{k=1}^{100} k \), you're calculating the sum of integers from 1 to 100.

The formula for the sum of the first \( n \) integers is \( \frac{n(n+1)}{2} \). Applying this formula simplifies the calculation. In the example with our summation problem \( \sum_{k=1}^{100} 3k \), each integer is multiplied by 3, a scaling factor, before summing.

Therefore, the process becomes:

• Determine the sum of integers using the formula.
• Multiply that sum by the scaling factor, 3 in this case.

This method reduces the complexity of computing large sums, making calculations more manageable and less error-prone.

constant term

A constant term in mathematics is a number on its own, or in other words, a term in an algebraic expression that does not change with the variable. In the expression \( 3k+4 \), the constant term is 4.

When evaluating summations like \( \sum_{k=1}^{100}(3k+4) \), it's important to isolate and sum the constant terms individually. Since the constant 4 appears for every value of \( k \) from 1 to 100, the sum of these constant terms is calculated by multiplying the constant by the number of times it appears:

• Given the expression \( 4 \) and that \( k \) ranges from 1 to 100, you calculate \( 4 \times 100 \).
• This results in \( 400 \), contributing to the overall summation.

Summing constant terms in this way simplifies the process by treating constants separately from variable terms, which ensures precise and straightforward calculations.