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Chapter 9: Problem 2

Fill in the blanks The ________ differences of a sequence are found by subtracting consecutive terms.

Short Answer

Expert verified
The correct term is 'successive'. The statement should read: The 'successive' differences of a sequence are found by subtracting consecutive terms.

Step by step solution

01

Understanding the concept

The blank in the statement refers to a method used in mathematical sequences. The operation involves subtracting consecutive terms in a sequence. This operation is referred to as finding the 'differences' of a sequence.
02

Filling the blank

When we subtract consecutive terms in a sequence, we are finding the 'successive', 'subsequent' or 'consecutive' differences. Here, the most fitting term that is typically used in mathematics is 'successive'. Therefore, the blank can be filled with the term 'successive'.

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Key Concepts

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

Successive Differences
Successive differences are an important concept in understanding mathematical sequences. They are found by taking two consecutive terms in a sequence and subtracting one from the other. This operation helps to determine patterns or rules that the sequence follows.

For example, if we have a sequence: 3, 7, 11, 15, the successive differences would be calculated as follows:
  • Subtract 3 from 7, which gives 4.
  • Subtract 7 from 11, which gives 4.
  • Subtract 11 from 15, which gives 4.
In this example, the successive differences are all the same, hinting at a regular pattern or interval in the sequence. Recognizing such patterns is essential for predicting future terms or understanding the structure of a sequence.
Consecutive Terms
The term 'consecutive terms' refers to terms in a sequence that follow one another in order, without any terms in between. These are directly next to each other in the sequence.

Understanding consecutive terms is crucial when finding successive differences because we always work with two consecutive terms for this operation.

Consider the simple sequence of even numbers: 2, 4, 6, 8. Here, 2 and 4 are consecutive, as are 4 and 6, and so on. Identifying consecutive terms helps us apply operations like subtraction to uncover broader patterns in sequences.
Mathematical Sequences
A mathematical sequence is an ordered list of numbers following a particular pattern or rule. Sequences can be finite or infinite. They are foundational in mathematics and are used to model various real-world phenomena.

Common types of sequences include arithmetic sequences, where each term is derived by adding a constant to the previous term, and geometric sequences, where each term is derived by multiplying the previous term by a constant.

For example, in the arithmetic sequence 5, 10, 15, 20, each term increases by 5, which is constant. Recognizing the type of sequence helps in predicting future terms and understanding the sequence's progression.

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Most popular questions from this chapter

In Exercises 79 - 86, solve for \( n \). \( _{n + 2} P_3 = 6 \cdot _{n + 2}P_1 \)

In Exercises 31 - 34, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and three red marbles. Both marbles are yellow.

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