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Chapter 5: Problem 136

What is an arithmetic sequence? Give an example with your description.

Short Answer

Expert verified
An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. In the sequence 3, 7, 11, 15, 19..., for instance, the common difference is 4.

Step by step solution

01

Defining an Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers such that the difference of any two successive members is a constant. This difference is referred to as the 'common difference.' The \( n^{th} \) term of an arithmetic sequence can be calculated using the formula \( a + (n-1)d \), where a is the first term, d is the common difference, and n is the term number.
02

Providing an Example

Let's consider an example. A sequence like 3, 7, 11, 15, 19, and so on, is an arithmetic sequence. Here, '3' is the first term (a), and the common difference (d) is '7-3' or '4'. To find, for instance, the 5th term, we substitute a = 3, d = 4 and n = 5 into the formula to get \( 3 + (5-1)4 = 19 \). Hence, the 5th term of this sequence is 19. Of course, in this case the 5th term is directly visible, but in case of long sequences or when the n is high, the formula comes in handy.

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Key Concepts

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

Common Difference
In arithmetic sequences, the term "common difference" is a fundamental concept. This difference is consistent between any two successive terms in the sequence.

For example, if you have the sequence 5, 8, 11, 14, each number increases by the same amount to get to the next. Here, the common difference is 3, which is calculated by subtracting the first term from the second (8 - 5 = 3).
  • The common difference can be positive, negative, or even zero. If it's positive, each term in the sequence will be larger than the one before it.
  • A negative common difference means the numbers decrease as the sequence progresses.
  • If the common difference is zero, all terms in the sequence are the same, leading to a constant series.

Identifying the common difference is key to understanding how the sequence grows or shrinks from one term to the next.
n-th Term Formula
The n-th term formula is a powerful tool used to find any term in an arithmetic sequence without listing all the preceding terms. The formula is expressed as: \[ a_n = a_1 + (n-1)d \]

Where:
  • \(a_n\) is the n-th term you're looking for.
  • \(a_1\) is the first term of the sequence.
  • \(d\) is the common difference.
  • \(n\) is the term number you want to find.
For example, in the sequence 3, 7, 11, 15, if you need the 10th term, you'd take \(a_1\) = 3, \(n\) = 10, and \(d\) = 4. Plug these into the formula: \[ 3 + (10-1) \times 4 = 3 + 36 = 39 \]

Thus, the 10th term of this sequence is 39. This formula is incredibly helpful for quickly finding terms in longer sequences, saving time and effort.
Arithmetic Progression
An arithmetic progression is another name for an arithmetic sequence. It describes a sequence of numbers with a constant difference between consecutive terms.

Using the example sequence 2, 6, 10, 14, the arithmetic progression clearly shows a uniform pattern thanks to the constant difference of 4.
  • Each step simply involves adding this common difference to the last term to get the next one.
  • The uniformity in this pattern makes arithmetic progression easy to identify and work with.

Arithmetic progressions are used in various mathematical applications, such as predicting patterns and solving for unknowns in problem-solving scenarios. Recognizing sequences that fit this pattern is crucial because it allows you to apply the n-th term formula effectively, solving problems more efficiently.

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 the term of the sequence. \(a_{1}=6, d=3\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=8000, r=-\frac{1}{2}\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(3,8,13,18, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=3\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-2, r=-3\)
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