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Chapter 12: Problem 48

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

Short Answer

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

Step by step solution

01

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 \(\sum_{k=1}^{100}(3k+4)\).
02

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 \(3k+4\).
03

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.
04

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 \(k=100\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

In this exercise we prove the identity $$ \left(\begin{array}{c}{n} \\\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{n+1} \\ {r}\end{array}\right) $$

Mortgage A couple secures a 30 -year loan of \(\$ 100,000\) at 9\(\frac{3}{4} \%\) per year, compounded monthly, to buy a house. (a) What is the amount of their monthly payment? (b) What total amount will they pay over the 30 -year period? (c) If, instead of taking the loan, the couple deposits the monthly payments in an account that pays 9\(\frac{3}{4} \%\) interest per year, compounded monthly, how much will be in the account at the end of the 30 -year period?

Find the second term in the expansion of $$ \left(x^{2}-\frac{1}{x}\right)^{25} $$

Mortgage What is the monthly payment on a 30 -year mortgage of \(\$ 100,000\) at 8\(\%\) interest per year, compounded monthly? What is the total amount paid on this loan over the 30 -year period?

Find the sum. $$ 1+3+9+\cdots+2187 $$
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