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Chapter 8: Problem 2

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the explicit formula \(a_{n}=1 / n,\) for \(n=1,2,3, \ldots .\) Write out the first five terms of the sequence.

Short Answer

Expert verified
Answer: The first five terms are \(1, 0.5, 0.333, 0.25, 0.2\).

Step by step solution

01

1. Find the first term of the sequence#a_1\(

To find the first term of the sequence, we'll substitute \)n=1\( into the formula \)a_n = \frac{1}{n}$. So: $$ a_1 = \frac{1}{1} = 1 $$
02

2. Find the second term of the sequence\(a_2\)

Now, we'll substitute \(n=2\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_2 = \frac{1}{2} = 0.5 $$
03

3. Find the third term of the sequence\(a_3\)

Next, we'll substitute \(n=3\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_3 = \frac{1}{3} \approx 0.333 $$
04

4. Find the fourth term of the sequence\(a_4\)

Now we'll substitute \(n=4\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_4 = \frac{1}{4} = 0.25 $$
05

5. Find the fifth term of the sequence\(a_5\)

Finally, we'll substitute \(n=5\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_5 = \frac{1}{5} = 0.2 $$
06

Result: First five terms of the sequence

Following the steps outlined above, we have found the first five terms of the sequence \(\left\\{a_{n}\right\\}\): $$ a_1 = 1, \quad a_2 = 0.5, \quad a_3 \approx 0.333, \quad a_4 = 0.25, \quad a_5 = 0.2 $$

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Most popular questions from this chapter

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=n^{1.001} \text { and } b_{n}=\ln n^{10}, n \geq 1$$

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{1}{n}=0$$

The Greek philosopher Zeno of Elea (who lived about 450 B.c.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued The slower when running will never be overtaken by the quicker: for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. In other words, by giving the tortoise a head start, Achilles will never overtake the tortoise because every time Achilles reaches the point where the tortoise was, the tortoise has moved ahead. Resolve this paradox by assuming that Achilles gives the tortoise a 1 -mi head start and runs \(5 \mathrm{mi} / \mathrm{hr}\) to the tortoise's \(1 \mathrm{mi} / \mathrm{hr}\). How far does Achilles run before he overtakes the tortoise, and how long does it take?

An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction \(p\) of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.
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