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Chapter 14: Problem 9

Find the first term and the common difference. $$ 2,6,10,14, \ldots $$

Short Answer

Expert verified
The first term is 2, and the common difference is 4.

Step by step solution

01

- Identify the First Term

The first term in a sequence is the initial number. In this sequence, the first term is given as 2.
02

- Identify the Common Difference

The common difference in an arithmetic sequence is found by subtracting the first term from the second term. Subtract 2 from 6 to find the common difference: \[6 - 2 = 4\].
03

- Verify the Common Difference

To ensure the common difference is consistent, subtract the second term from the third term: \[10 - 6 = 4\]. Continue by subtracting the third term from the fourth term: \[14 - 10 = 4\]. The common difference is indeed constant and equal to 4.

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Key Concepts

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

Understanding the First Term
The first term of an arithmetic sequence is simply the initial number in the sequence. This is the starting point from which all subsequent terms are derived. In our given sequence, \[2, 6, 10, 14, \ldots\]{2,6,10,14, \ldots\}, the first term is 2. It's crucial because it sets the foundation for finding every other term in the sequence.
Identifying the Common Difference
The common difference is the amount by which each term increases to get to the next term in an arithmetic sequence. You find it by subtracting the first term from the second term. For instance, in our sequence \[2, 6, 10, 14, ...\], subtracting 2 from 6 gives \[6 - 2 = 4\]. This 4 is the common difference.
You can check this by looking at the other consecutive terms:
  • \[10 - 6 = 4\]
  • \[14 - 10 = 4\]

Each time, the common difference is consistent, verifying its accuracy.
Sequence Verification
Verification ensures our calculated common difference holds true throughout the sequence.
To verify the common difference in our sequence
  • First pair: \[6 - 2 = 4\]
  • Second pair: \[10 - 6 = 4\]
  • Third pair: \[14 - 10 = 4\]

All of these consistent subtractions confirm that the common difference is indeed 4. This verification step is essential for confirming our results.

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