Problem 18
Question
Decide whether each sequence is finite or infinite. $$-1,-2,-3,-4$$
Step-by-Step Solution
Verified Answer
The sequence is finite with exactly four terms.
1Step 1: Identify if there's a pattern
Look at the given sequence: \(-1, -2, -3, -4\). The numbers decrease by 1 each time. This indicates a consistent pattern in the sequence perhaps forming an arithmetic sequence with a common difference of -1.
2Step 2: Check the listed elements
Consider the sequence of numbers provided: \(-1, -2, -3, -4\). Examine if all the terms in this sequence have been listed or if there is an atomic or closing term. Here, only four specific terms are given, and there‘s no suggestion of continuation.
3Step 3: Determine if there are more terms
Determine if you can generate more numbers following the given pattern. In this sequence, since there is no indication of continuation (like an ellipsis "..."), assume these are the only values present.
4Step 4: Decide if it's finite or infinite
Based on the observation that the sequence contains only the numbers \(-1, -2, -3, -4\) and ends without indication of further numbers or ellipsis, we conclude that this sequence has a finite number of four terms.
Key Concepts
Arithmetic SequenceCommon DifferencePattern RecognitionMathematical Sequences
Arithmetic Sequence
An arithmetic sequence is a series of numbers in which the difference between any two consecutive numbers is constant. This difference is known as the "common difference". In simpler terms, to form an arithmetic sequence, you add (or subtract) the same number repeatedly to reach the next term in the sequence. This leads to a linear pattern.
For instance, consider the sequence \(-1, -2, -3, -4\). Observe how each number follows the previous by subtracting 1, indicating that \(-1\) is our common difference. By following this rule, you know exactly what the next number would be if the sequence were to continue.
For instance, consider the sequence \(-1, -2, -3, -4\). Observe how each number follows the previous by subtracting 1, indicating that \(-1\) is our common difference. By following this rule, you know exactly what the next number would be if the sequence were to continue.
Common Difference
The term "common difference" is the backbone of an arithmetic sequence. It is the constant value that differentiates one term from the next.
In our sequence example \(-1, -2, -3, -4\), the common difference is \(-1\), because each number is 1 less than the number preceding it. It's not just a number; it's a key indicator of the pattern.
In our sequence example \(-1, -2, -3, -4\), the common difference is \(-1\), because each number is 1 less than the number preceding it. It's not just a number; it's a key indicator of the pattern.
- The common difference helps us determine the trajectory of the sequence.
- It allows us to predict future terms if the sequence continued.
- Recognizing this constant can help identify if a sequence is arithmetic.
Pattern Recognition
Pattern recognition involves identifying the rules that dictate the progression in a sequence. By examining patterns, we can decode how a sequence builds upon itself.
With \(-1, -2, -3, -4\), you recognize a simple pattern where each subsequent number is decreased by 1. This recognition is what leads us to identify it as an arithmetic sequence with predictable behavior.
With \(-1, -2, -3, -4\), you recognize a simple pattern where each subsequent number is decreased by 1. This recognition is what leads us to identify it as an arithmetic sequence with predictable behavior.
- Pattern recognition is key to understanding sequences and series.
- Recognizing a pattern can help you quickly determine things like the next numbers or how sequences are structured.
- It establishes whether a sequence is finite or infinite by understanding its continuation or termination points.
Mathematical Sequences
Mathematical sequences are ordered lists of numbers following specific rules. These sequences can either be finite or infinite.
A finite sequence has a distinct start and end point, just like our example \(-1, -2, -3, -4\). An infinite sequence continues indefinitely without terminating.
Sequences can take many forms, with arithmetic and geometric being common categories. The crucial part of sequences is understanding how each number relates to its predecessor.
A finite sequence has a distinct start and end point, just like our example \(-1, -2, -3, -4\). An infinite sequence continues indefinitely without terminating.
Sequences can take many forms, with arithmetic and geometric being common categories. The crucial part of sequences is understanding how each number relates to its predecessor.
- Finite sequences conclude after a set number of terms.
- Infinite sequences go on infinitely, often indicated by an ellipsis (...).
- Mathematical sequences help in understanding progressions and series, aiding various fields like finance, computations, and more.
Other exercises in this chapter
Problem 18
Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{c}n \\\n-2\end{array}\right)$$
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Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. $$a_{n}>\frac{1}{10}, \text { where } a_{n}=\fr
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Work each problem. A woman invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If any one guest is as likely to arrive first as an
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Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$(1+x)^{n} \geq 1+n x, \text { for } x>-1$$
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