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Chapter 11: Problem 12

Find the first term and the common difference. $$\$ 825, \$ 804, \$ 783, \$ 762, \dots$$

Short Answer

Expert verified
The first term is $825, and the common difference is -21.

Step by step solution

01

Identify the Arithmetic Sequence

Recognize that the given sequence is an arithmetic sequence, where each term after the first is obtained by adding a constant difference.
02

Find the Common Difference

Calculate the common difference (\(d\)) by subtracting the second term from the first term. So, \(d = 804 - 825 = -21\). This means the common difference is -21.
03

Find the First Term

The first term (\(a_1\)) is the initial term of the given sequence. Here, it is clearly stated as \$ 825.
04

Verify the Common Difference

Subtract subsequent terms to verify the common difference. \(804 - 825 = -21\), \(783 - 804 = -21\), and \(762 - 783 = -21\). Consistency in results confirms the common difference is -21.

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Key Concepts

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

Common Difference
In an arithmetic sequence, each term is found by adding or subtracting a fixed number called the 'common difference'. Understanding this concept is key to mastering arithmetic sequences.
The common difference (\(d\)) can be found by subtracting any term from the term that follows it.
  • For example, in the sequence given: \$ 825, \$ 804, \$ 783, \$ 762
  • The common difference is found by calculating: \(804 - 825 = -21\).
  • Similarly, for the next two terms, \(783 - 804 = -21\) and \(762 - 783 = -21\).
Here, the common difference is consistently -21, showing that this is indeed an arithmetic sequence where each term decreases by 21.
First Term
The first term of an arithmetic sequence is the starting point of the sequence. It is typically denoted as \(a_1\).
Identifying the first term is the first step in understanding an arithmetic sequence.
In your given sequence: \$ 825, \$ 804, \$ 783, \$ 762
The first term is \$ 825.
This value sets the stage for generating the rest of the sequence by consistently adding the common difference. For instance, starting from \$ 825 and subtracting 21 (the common difference) gives the next term, \$ 804. This process is repeated to form the rest of the sequence.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is known as the 'common difference'.
In more formal terms, if \(a_1\) is the first term and \(d\) is the common difference, then the \(n\)-th term (\(a_n\)) of the sequence can be expressed as:
\[ a_n = a_1 + (n-1)d \]
  • For example, with \$ 825 as \(a_1\) and -21 as \(d\), the sequence can be computed as follows:
  • Second term: \(a_2 = 825 + (2-1)(-21) = 825 - 21 = 804\)
  • Third term: \(a_3 = 825 + (3-1)(-21) = 825 - 42 = 783\)
  • Fourth term: \(a_4 = 825 + (4-1)(-21) = 825 - 63 = 762\)
Understanding the formula for arithmetic sequences helps in quickly identifying or generating any term in the sequence.

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