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Chapter 14: Problem 11

Find the first term and the common difference. $$ 7,3,-1,-5, \ldots $$

Short Answer

Expert verified
First term: 7, Common difference: -4.

Step by step solution

01

Identify the first term

The first term of the sequence is the initial number. Here, the first term is 7.
02

Determine the common difference

The common difference is found by subtracting the first term from the second term. In this case, \(3 - 7 = -4\). To verify, check the difference between subsequent terms: \(-1 - 3 = -4\) and \(-5 - (-1) = -4\). Thus, the common difference is -4.

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Key Concepts

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

First Term
In arithmetic sequences, the first term is crucial. It sets the stage for all following terms. The first term, denoted usually as \(a_1\), is the initial number in the sequence.
In our example:
7, 3, -1, -5, ...
The first term is 7.
This means the sequence starts with 7.
Once the first term is defined, it allows us to systematically calculate the subsequent terms using the common difference.
Common Difference
The common difference is essential for arithmetic sequences as it represents the constant interval between successive terms. To find it, subtract the first term (a1) from the second term (a2).
For example:
In the given sequence 7, 3, -1, -5, ... we calculate:
\(3 - 7 = -4\)
To ensure consistency, check the difference between other consecutive terms:
  • \(-1 - 3 = -4\)
  • \(-5 - (-1) = -4\)
Therefore, the common difference, denoted usually as \(d\), is -4.
Understanding the common difference helps in predicting and computing future terms.
Subsequent Terms
After identifying the first term and the common difference, you can easily determine the subsequent terms. Each term in an arithmetic sequence is obtained by adding the common difference to the previous term.
Using our sequence 7, 3, -1, -5, ... and the common difference of -4:
  • First term, \(a_1 = 7\)
  • Second term, \(a_2 = 7 + (-4) = 3\)
  • Third term, \(a_3 = 3 + (-4) = -1\)
  • Fourth term, \(a_4 = -1 + (-4) = -5\)
This pattern continues indefinitely.
Knowing how to calculate subsequent terms is helpful for solving problems and understanding the behavior of arithmetic sequences.

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