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Chapter 14: Problem 99

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Short Answer

Expert verified
The differences \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},a_{5}-a_{4}\) are all equal to 4, indicating that the sequence is an arithmetic sequence with a common difference of 4.

Step by step solution

01

Find the First Five Terms of the Sequence

Use the given formula to compute the first five terms of the sequence: \(a_{1} = 4*1 - 3 = 1\),\(a_{2} = 4*2 - 3 = 5\),\(a_{3} = 4*3 - 3 = 9\),\(a_{4} = 4*4 - 3 = 13\),\(a_{5} = 4*5 - 3 = 17\).
02

Calculate the Differences

Take the differences of each consecutive term:\(a_{2}-a_{1} = 5 - 1 = 4\),\(a_{3}-a_{2} = 9 - 5 = 4\),\(a_{4}-a_{3} = 13 - 9 = 4\),\(a_{5}-a_{4} = 17 - 13 = 4\).
03

Observations

Upon calculating the difference between the consecutive terms, it is observed that all the differences are equal to 4. This confirms that the given sequence is an arithmetic sequence, where the common difference is 4.

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