Problem 2
Question
Fill in the blanks The ________ differences of a sequence are found by subtracting consecutive terms.
Step-by-Step Solution
Verified Answer
The correct term is 'successive'. The statement should read: The 'successive' differences of a sequence are found by subtracting consecutive terms.
1Step 1: 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.
2Step 2: 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'.
Key Concepts
Successive DifferencesConsecutive TermsMathematical Sequences
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:
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.
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.
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.
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.
Other exercises in this chapter
Problem 2
In Exercises 1 - 7, fill in the blanks. The set of all possible outcomes of an experiment is called the ________ ________.
View solution Problem 2
Fill in the blanks An ordering of \( n \) elements is called a ________ of the elements.
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Fill in the blanks. The \( n \)th term of an arithmetic sequence has the form ________.
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Fill in the blanks. The function values \( a_1, a_2, a_3, a_4, \cdots \) are called the ________ of a sequence.
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