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Chapter 2: Problem 5

use inductive reasoning to find the next two terms in each sequence. 7,3,-1,-5,-9,-13

Short Answer

Expert verified
-17, -21

Step by step solution

01

- Identify the pattern

First, observe the difference between consecutive terms in the sequence. From 7 to 3, the difference is 3 - 7 = -4. From 3 to -1, the difference is -1 - 3 = -4. Continue checking if the difference remains consistent.
02

- Confirm the consistent difference

Next, check further. From -1 to -5, the difference is -5 - (-1) = -4. From -5 to -9, the difference is -9 - (-5) = -4. Finally, from -9 to -13, the difference is -13 - (-9) = -4. This confirms that the sequence decreases by 4 consistently.
03

- Calculate the next term

To find the next term, subtract 4 from the last term in the sequence. So, -13 - 4 = -17.
04

- Calculate the next term after that

Repeat the same process. Subtract 4 from the last calculated term: -17 - 4 = -21.
05

- State the next two terms

The next two terms in the sequence are -17 and -21.

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Key Concepts

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

Sequence Pattern
A sequence is a list of numbers arranged in a specific order. The challenge is to find the rule that connects each number to the next. This rule is called the pattern. By identifying this pattern, you can predict the following numbers in the sequence.

Consider the sequence: 7, 3, -1, -5, -9, -13. To find the pattern, look at how each number changes to the next:
  • From 7 to 3
  • From 3 to -1
  • From -1 to -5
  • From -5 to -9
  • From -9 to -13
Notice how you subtract 4 each time. This is your pattern.
Understanding the pattern helps to determine what comes next by simply repeating the same rule.
Arithmetic Sequence
An arithmetic sequence is a type of sequence where the difference between consecutive terms is always the same. This common difference can be positive, negative, or even zero.

In the sequence 7, 3, -1, -5, -9, -13, we see that each term is obtained by subtracting 4 from the previous term.
This means it's an arithmetic sequence with a common difference of -4. Detecting an arithmetic sequence involves:
  • Calculating the difference between consecutive terms
  • Ensuring the difference remains consistent
Once you confirm a common difference, you can use it to generate any number of future terms by continuing to add (or subtract) that difference.
Finding Next Terms
To find the next terms in an arithmetic sequence, apply the common difference to the last known term repeatedly.

Let's use the sequence: 7, 3, -1, -5, -9, -13. We established the common difference is -4. To find the terms after -13, follow these steps:
  • Subtract 4 from -13: \(-13 - 4 = -17\)
  • Subtract 4 from -17: \(-17 - 4 = -21\)
So, the next two terms are -17 and -21.
Repeating this process allows you to keep extending the sequence indefinitely. It's all about using that consistent common difference.

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Most popular questions from this chapter

The definition of a parallelogram says, "If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram" Quadrilateral LNDA has both pairs of opposite sides parallel. What conclusion can you make? What type of reasoning did you use?

Use your ruler and protractor to draw a triangle with angles measuring \(40^{\circ}\) and \(60^{\circ}\) and a side between them with length \(9 \mathrm{cm}\). Explain your method. Can you draw a second triangle using the same instructions that is not congruent to the first?

If two congruent angles are supplementary, what must be true of the two angles? Make a sketch, then complete the following conjecture: If two angles are both congruent and supplementary, then ____________________ .

Think of a situation you have observed in which inductive reasoning was used incorrectly. Write a paragraph or two describing what happened and explaining why you think it was an incorrect use of inductive reasoning.

Use inductive reasoning to find the next two terms of the sequence. $$\frac{1}{2}, 9, \frac{2}{3}, 10, \frac{3}{4}, 11,1,?,?$$
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