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Chapter 9: Problem 6

Determine whether the sequence is arithmetic. If so, then find the common difference. $$4,9,14,19,24, . . .$$

Short Answer

Expert verified
The given sequence is an arithmetic sequence with a common difference of 5.

Step by step solution

01

Identifying the Type of Sequence

To determine whether this sequence is arithmetic, compare the differences between consecutive terms. For the sequence given \(4, 9, 14, 19, 24,...\) the differences are \(9 - 4 = 5\), \(14 - 9 = 5\), \(19 - 14 = 5\), and \(24 - 19 = 5\). The differences between each pair of consecutive terms are all the same, so this indicates that the sequence is arithmetic.
02

Finding the Common Difference

The difference between each consecutive pair of terms within an arithmetic sequence remains constant; this difference is known as the common difference. So here, the common difference is 5.
03

Final validation

The sequence is arithmetic with a common difference of 5, as the difference between every two consecutive terms is constantly 5.

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Key Concepts

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

Understanding Common Difference in Arithmetic Sequences
In an arithmetic sequence, the common difference is a crucial component. It is the fixed value you add (or subtract, if negative) to move from one term to the next. This consistency in the gap between terms is what defines an arithmetic sequence. For example, take the sequence provided: \(4, 9, 14, 19, 24, \ldots\).
  • The first step is calculating the differences: \(9 - 4 = 5\), \(14 - 9 = 5\), \(19 - 14 = 5\), and \(24 - 19 = 5\).
  • Since these differences are equal, we can confirm that the pattern of adding \(5\) persists throughout, which solidifies our sequence as arithmetic.
  • Thus, the common difference here is \(5\).
This implies each term increases by \(5\) compared to the previous one. Understanding and finding the common difference is often the first step in working with arithmetic sequences!
Exploring Consecutive Terms
Consecutive terms in a sequence are terms that follow one right after another. They are essential in understanding the pattern of any arithmetic sequence.
  • In the sequence \(4, 9, 14, 19, 24, \ldots\), each number is a consecutive term.
  • Identifying consecutive terms helps in confirming the nature of the sequence. By calculating the differences (as previously shown), we see the relationship between these terms.
In arithmetic sequences, this consistent step between consecutive terms doesn't vary; it is precisely this nature that makes an arithmetic pattern identifiable and straightforward to model. By connecting each term through the common difference, you can also predict subsequent values or even solve for unknowns within the sequence!
Introduction to Arithmetic Series
An arithmetic series pertains to the sum of terms in an arithmetic sequence. While the sequence shows individual terms lined up with a common difference, the series adds these terms together.
  • For instance, if our sequence were finite, like \(4, 9, 14, 19, 24\), the arithmetic series would be the sum: \(4 + 9 + 14 + 19 + 24\).
  • Calculating an arithmetic series involves simple addition when dealing with only a few terms, but more extended series can use the formula: \(S_n = \frac{n}{2} (a + l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.
Understanding an arithmetic series helps in various applications, such as economics or day-to-day calculations where predictions and sum calculations are needed. It enhances your understanding of sequences by broadening thinking from individual terms to their cumulative effect.

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