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Chapter 11: Problem 11

Write the first five terms of each sequence. $$a_{n}=\frac{n^{3}+8}{n+2}$$

Short Answer

Expert verified
3, 4, 7, 12, 19

Step by step solution

01

- Substitute n=1

Replace the variable n with 1 in the given expression: \(a_1 = \frac{1^3 + 8}{1 + 2} = \frac{1 + 8}{3} = \frac{9}{3} = 3\). Thus, the first term is 3.
02

- Substitute n=2

Replace the variable n with 2: \(a_2 = \frac{2^3 + 8}{2 + 2} = \frac{8 + 8}{4} = \frac{16}{4} = 4\). Thus, the second term is 4.
03

- Substitute n=3

Replace the variable n with 3: \(a_3 = \frac{3^3 + 8}{3 + 2} = \frac{27 + 8}{5} = \frac{35}{5} = 7\). Thus, the third term is 7.
04

- Substitute n=4

Replace the variable n with 4: \(a_4 = \frac{4^3 + 8}{4 + 2} = \frac{64 + 8}{6} = \frac{72}{6} = 12\). Thus, the fourth term is 12.
05

- Substitute n=5

Replace the variable n with 5: \(a_5 = \frac{5^3 + 8}{5 + 2} = \frac{125 + 8}{7} = \frac{133}{7} = 19\). Thus, the fifth term is 19.

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Key Concepts

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

sequence terms
When working with sequences, it is important to understand what 'sequence terms' refer to. A sequence is simply a list of numbers in a specific order, and each number in the list is called a term. In the given exercise, we are identifying the first five terms of a sequence defined by the formula: $$a_{n}=\frac{n^{3}+8}{n+2}$$. By substituting different values of n (such as 1, 2, 3, 4, and 5), we can find the corresponding terms. Remember that the position of a term in a sequence, such as the first term, second term, etc., is essential for understanding and identifying patterns or relationships between the terms.
algebraic expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the sequence we are examining, $$a_{n}=\frac{n^{3}+8}{n+2}$$, the expression involves both addition and division operations. The variable n represents the term's position in the sequence. To evaluate the expression, we need to correctly perform these operations. Breaking down complex algebraic expressions step-by-step helps in avoiding errors and simplifies the sequence evaluation.
variable substitution
Variable substitution is an essential technique in evaluating functions or expressions. It involves replacing the variable with a specific value. For example, in our sequence formula, substitute n with integers 1, 2, 3, 4, and 5. This creates different instances of the expression, such as $$a_1 = \frac{1^3 + 8}{1 + 2}$$ and $$a_2 = \frac{2^3 + 8}{2 + 2}$$. Be meticulous about substituting the variables and performing arithmetic operations accurately to solve the equations.
function evaluation
Function evaluation is the process of calculating the output of a function for a specific input. In the context of our exercise, the function is $$a_{n}=\frac{n^{3}+8}{n+2}$$. To evaluate the function, we use variable substitution for different values of n and simplify the result. This step-by-step evaluation reveals the terms of the sequence: 3, 4, 7, 12, and 19 for n = 1, 2, 3, 4, and 5, respectively. This process helps us understand how varying the input (n) affects the output (the sequence term).

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Most popular questions from this chapter

Find the sum of the first six terms of the series $$\frac{\pi^{4}}{90}=\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\frac{1}{5^{4}}+\cdots+\frac{1}{n^{4}}+\cdots$$ Multiply this result by \(90,\) and take the fourth root to obtain an approximation of \(\pi\). Compare your answer to the actual decimal approximation of \(\pi\).

Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, 1)=n$$

The recursively defined sequence \(a_{1}=k\) $$ a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right), \quad \text { if } n>1$$ can be used to compute \(\sqrt{k}\) for any positive number \(k\). This sequence was known to Sumerian mathematicians 4000 years ago, and it is still used today. Use this sequence to approximate the given square root by finding \(a_{6} .\) Compare your result with the actual value. (Source: Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.) (a) \(\sqrt{2}\) (b) \(\sqrt{11}\)

A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to an urban convention. (a) How many delegations are possible? (b) How many delegations could have all liberals? (c) How many delegations could have 2 liberals and 1 conservative? (d) If 1 member of the council serves as mayor, how many delegations are possible that include the mayor?

You are offered a 6 -week summer job and are asked to select one of the following salary options. Option I: 5000 dollars for the first day with a 10,000 dollars raise each day for the remaining 29 days (that is, 15,000 dollars for day 2,25,000 dollars for day \(3,\) and so on) Option 2: 0.01 dollars for the first day with the pay doubled each day (that is, 0.02 dollars for day \(2,\) 0.04 dollars for day 3, and so on. Which option would you choose?
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