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Chapter 11: Problem 55

Use the summation feature of a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{j=3}^{8} 2(0.4)^{j}$$

Short Answer

Expert verified
0.212

Step by step solution

01

Identify the Components of the Sum

The sum given is \(\sum_{j=3}^{8} 2(0.4)^{j}\). This means the summation starts at \(j = 3\) and ends at \(j = 8\). The expression to be summed is \(2(0.4)^{j}\).
02

Input the Summation into the Graphing Calculator

Using a graphing calculator's summation function, input the starting point \(j = 3\), the ending point \(j = 8\), and the expression \(2(0.4)^{j}\).
03

Calculate Each Term

Individually calculate each term of the summation: \(2(0.4)^{3}, \ 2(0.4)^{4}, \ 2(0.4)^{5}, \ 2(0.4)^{6}, \ 2(0.4)^{7}, \ 2(0.4)^{8}\).
04

Sum the Results

Add each calculated term to obtain the total sum: \(\sum_{j=3}^{8} 2(0.4)^{j} = 0.128 + 0.0512 + 0.02048 + 0.008192 + 0.0032768 + 0.00131072\).
05

Round the Sum to the Nearest Thousandth

Calculate the total sum and round to the nearest thousandth. The total sum is \(0.128 + 0.0512 + 0.02048 + 0.008192 + 0.0032768 + 0.00131072 \approx 0.212\).

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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 can be an extremely useful tool for evaluating sums like the one given in the exercise. To input the summation feature, you'll follow these steps:
  • Access the summation notation on your calculator (typically found under the 'math' menu).

  • Input the lower limit of your sum (in this case, \(j = 3\)).

  • Input the upper limit of your sum (in this case, \(j = 8\)).

  • Input the expression to be summed, \(2(0.4)^{j}\).

Once you've entered all the necessary components, the graphing calculator will automatically compute each term and provide the total sum. This saves time and reduces errors compared to manual calculations.
Using a graphing calculator is particularly helpful when dealing with more complex functions or larger numbers of terms.
exponential function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the exercise, the exponential function is \((0.4)^{j}\).
  • Here, 0.4 is the base and \(j\) is the exponent.

  • The exponent \(j\) changes from 3 to 8 in this specific summation.

Exponential functions are common in many fields, such as finance for compound interest and biology for population growth.
They're known for their rapid growth or decay rate, depending on whether the base is greater than or less than 1.
In this case, because the base (0.4) is less than 1, the value of \((0.4)^{j}\) becomes smaller as \(j\) increases.
rounding
Rounding is the process of reducing the number of significant digits in a number. It's often used to make results easier to read or to meet specific criteria, like rounding to the nearest thousandth in this exercise. Here's how to do it:
  • Identify the place value to which you want to round. The thousandth place is three decimal places to the right of the decimal point.

  • Look one digit to the right of the target place value. If that digit is 5 or greater, you round up the target place value by one. If it's less than 5, you leave the target place value as it is.

In the exercise, the unrounded sum is approximately 0.21218272. When rounding to the nearest thousandth, you look at the fourth digit to the right of the decimal point, which is 1 in this case.
Since 1 is less than 5, you leave the third digit (2) unchanged. Thus, the rounded sum is 0.212.

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Most popular questions from this chapter

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=2 e^{n}$$

The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

From a pool of 7 secretaries, 3 are selected to be assigned to 3 managers, 1 secretary to each manager. In how many ways can this be done?

You are offered a 6 -week summer job and are asked to select one of the following salary options. Option I: 5000 dollars for the first day with a 10,000 dollars raise each day for the remaining 29 days (that is, 15,000 dollars for day 2,25,000 dollars for day \(3,\) and so on) Option 2: 0.01 dollars for the first day with the pay doubled each day (that is, 0.02 dollars for day \(2,\) 0.04 dollars for day 3, and so on. Which option would you choose?

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$
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