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Chapter 5: Problem 147

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sequence \(1,4,8,13,19,26, \ldots\) is an arithmetic sequence.

Short Answer

Expert verified
The statement 'The sequence \(1,4,8,13,19,26, \ldots\) is an arithmetic sequence' is false. The correct version would be: 'The sequence \(1,4,8,13,19,26, \ldots\) is not an arithmetic sequence.'

Step by step solution

01

Calculate differences between consecutive terms.

Take the second term and subtract the first term, then the third term and subtract the second term, and so on until the last given term. This yields the sequence of differences \(4-1, 8-4, 13-8, 19-13, 26-19\) which simplifies to \(3, 4, 5, 6, 7\).
02

Compare differences.

Compare the differences obtained from Step 1. In this case, the differences are \(3, 4, 5, 6, 7\) which are not the same.
03

Make conclusions and amend statement if necessary.

As the differences between consecutive terms are not the same, the sequence is not an arithmetic sequence. To correct the statement, it could be amended to 'The sequence \(1,4,8,13,19,26, \ldots\) is not an arithmetic sequence.'

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