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Chapter 14: Problem 9

Write the first four terms of each sequence whose general term is given. $$a_{n}=\frac{2 n}{n+4}$$

Short Answer

Expert verified
The first four terms of the sequence are: \( \frac{2}{5}, \frac{2}{3}, \frac{6}{7}, 1 \)

Step by step solution

01

Understanding the sequence's general term

The sequence given is defined by a general term \(a_{n}=\frac{2n}{n+4}\). This means that for each natural number 'n', the nth term of this sequence can be found by substituting 'n' into this equation.
02

Substituting values for n

To find the first four terms of the sequence, we substitute \(n = 1, 2, 3, 4\) into the general term \(a_{n}\). This gives us the following: \n - For \(n=1\), \(a_{1}=\frac{2(1)}{1+4} = \frac{2}{5}\) \n - For \(n=2\), \(a_{2}=\frac{2(2)}{2+4} = \frac{4}{6} = \frac{2}{3}\) (simplified) \n - For \(n=3\), \(a_{3}=\frac{2(3)}{3+4} = \frac{6}{7}\) \n - For \(n=4\), \(a_{4}=\frac{2(4)}{4+4} = \frac{8}{8} = 1 (simplified)

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Most popular questions from this chapter

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$-1,1,-1,1, \dots$$

Explain how to find the general term of a geometric sequence.

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{100}{n} ; n:[0,1000,100] \text { by } a_{n}:[0,1,0.1]$$

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.
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