91Ó°ÊÓ

Exercises 43-46 describe procedures that are to be applied to numbers. In each exercise, a. Repeat the procedure for four numbers of your choice. Write a conjecture that relates the result of the process to the original number selected. b. Use the variable n to represent the original number and use deductive reasoning to prove the conjecture in part (a). Select a number. Add 5. Double the result. Subtract 4 . Divide by 2 . Subtract the original selected number.

In Exercises 47-52, use inductive reasoning to predict the next line in each sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct \(1 \times 9-1=8\) \(21 \times 9-1=188\) \(321 \times 9-1=2888\) \(4321 \times 9-1=38,888\)

Explain how to round \(14.26841\) to the nearest hundredth and to the nearest thousandth.

What does the \(\approx\) symbol mean?

In Exercises 57-60, identify the reasoning process, induction or deduction, in each example. Explain your answer. It can be shown that $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$ I can use this formula to conclude that the sum of the first one hundred counting numbers, \(1+2+3+\cdots+100\), is $$ \frac{100(100+1)}{2}=\frac{100(101)}{2}=50(101) \text {, or } 5050 \text {. } $$

The course policy states that work turned in late will be marked down a grade. I turned in my report a day late, so it was marked down from \(\mathrm{B}\) to \(\mathrm{C}\)

The ancient Greeks studied figurate numbers, so named because of their representations as geometric arrangements of points. a. Use inductive reasoning to write the five triangular numbers that follow \(21 .\) b. Use inductive reasoning to write the five square numbers that follow \(25 .\) c. Use inductive reasoning to write the five pentagonal numbers that follow 22 . d. Use inductive reasoning to complete this statement: If a triangular number is multiplied by 8 and then 1 is added to the product, a number is obtained.

You are on vacation in an isolated town. Everyone in the town was born there and has never left. You develop a toothache and check out the two dentists in town. One dentist has gorgeous teeth and one has teeth that show the effects of poor dental work. Which dentist should you choose and why?

India Jones is standing on a large rock in the middle of a square pool filled with hungry, man-eating piranhas. The edge of the pool is 20 feet away from the rock. India's mom wants to rescue her son, but she is standing on the edge of the pool with only two planks, each \(19 \frac{1}{2}\) feet long. How can India be rescued using the two planks?

(This logic problem dates back to the eighth century.) A farmer needs to take his goat, wolf, and cabbage across a stream. His boat can hold him and one other passenger (the goat, wolf, or cabbage). If he takes the wolf with him, the goat will eat the cabbage. If he takes the cabbage, the wolf will eat the goat. Only when the farmer is present are the cabbage and goat safe from their respective predators. How does the farmer get everything across the stream?