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Chapter 8: Problem 16

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) $$a_{n}=2 n^{2}-3 n+1$$

Short Answer

Expert verified
Answer: The first four terms of the sequence are \(0\), \(3\), \(10\), and \(21\).

Step by step solution

01

Find the term \(a_1\)

To find the first term of the sequence, replace \(n\) by \(1\) in the given formula: $$a_1 = 2(1)^2 - 3(1) + 1$$ Evaluating this expression, we get: $$a_1 = 2 - 3 + 1 = 0$$
02

Find the term \(a_2\)

To find the second term of the sequence, replace \(n\) by \(2\) in the given formula: $$a_2 = 2(2)^2 - 3(2) + 1$$ Evaluating this expression, we get: $$a_2 = 8 - 6 + 1 = 3$$
03

Find the term \(a_3\)

To find the third term of the sequence, replace \(n\) by \(3\) in the given formula: $$a_3 = 2(3)^2 - 3(3) + 1$$ Evaluating this expression, we get: $$a_3 = 18 - 9 + 1 = 10$$
04

Find the term \(a_4\)

To find the fourth term of the sequence, replace \(n\) by \(4\) in the given formula: $$a_4 = 2(4)^2 - 3(4) + 1$$ Evaluating this expression, we get: $$a_4 = 32 - 12 + 1 = 21$$
05

Write the first four terms of the sequence

We found the first four terms of the sequence to be \(0\), \(3\), \(10\), and \(21\). Therefore, the first four terms of the sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are: $$a_1 = 0, \ a_2 = 3, \ a_3 = 10, \ a_4 = 21$$

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a list of numbers with a common difference between them. This means that when we move from one term to the next, we either add or subtract a fixed amount. For example, in the sequence 2, 5, 8, 11, the common difference is 3 because we're adding 3 to each term to get the next one.

One important point to understand is how to determine the nth term of an arithmetic sequence, which can be calculated using the formula:
\[ a_n = a_1 + (n - 1) \times d \]
Here, \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) represents the position of the term in the sequence, and \( d \) is the common difference. This formula allows us to find any term in an arithmetic sequence without listing out all the preceding terms.
Quadratic Sequence
On the other hand, a quadratic sequence is a sequence of numbers where the second differences between terms are constant. Unlike the arithmetic sequence where the first differences are constant, a quadratic sequence shows that as we move from one term to the next, the increase (or decrease) is not at a consistent pace but rather increases (or decreases) at a consistent rate over time.

For instance, given the terms 0, 3, 10, 21, the first differences are 3, 7, and 11, while the second differences are 4 and 4, a constant. This indicates that the sequence is quadratic. A quadratic sequence can be represented algebraically as:
\[ a_n = a \times n^2 + b \times n + c \]
where \( a_n \) is the nth term, and \( a \), \( b \), and \( c \) are constants that define the properties of the quadratic sequence.
Sequence Term Formula
Understanding the sequence term formula is crucial for identifying patterns in sequences and predicting future terms. Depending on the sequence type, different formulas are applied.

For the arithmetic sequence, the formula we earlier discussed is used. However, for a quadratic sequence, as in the exercise considered, the nth term is found using a quadratic formula like the one given, \( a_n = 2n^2 - 3n + 1 \). These formulas are algebraic expressions that depend on the value of \( n \), the term's position in the sequence, which helps to compute the value of any term without having to calculate all previous terms.
Algebraic Expressions
Algebraic expressions, such as the ones used to describe sequence terms, are combinations of constants, variables (like \( n \) in the sequences), and arithmetic operations (addition, subtraction, multiplication, and sometimes division). They are the building blocks of algebra and provide a way to represent mathematical relationships and patterns.

For example, in the exercise, the expression \( 2n^2 - 3n + 1 \) is an algebraic expression that defines the pattern of the quadratic sequence. By plugging in different values of \( n \), we can calculate different terms of the sequence. The beauty of algebraic expressions is that they paint a clear mathematical picture that can be manipulated to solve for different variables under various conditions.

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Most popular questions from this chapter

For a positive real number \(p,\) the tower of exponents \(p^{p^{p}}\) continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence \(\left\\{p^{p},\left(p^{p}\right)^{p},\left(\left(p^{p}\right)^{p}\right)^{p}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{1}=p^{p} .\) The tower could also be built from the bottom as the limit of the sequence \(\left\\{p^{p}, p^{\left(p^{p}\right)}, p^{\left(p^{(i)}\right)}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=p^{a_{n}}(\text { building from the bottom })\) where again \(a_{1}=p^{p}\). a. Estimate the value of the tower with \(p=0.5\) by building from the top. That is, use tables to estimate the limit of the sequence defined recursively by (1) with \(p=0.5 .\) Estimate the maximum value of \(p > 0\) for which the sequence has a limit. b. Estimate the value of the tower with \(p=1.2\) by building from the bottom. That is, use tables to estimate the limit of the sequence defined recursively by (2) with \(p=1.2 .\) Estimate the maximum value of \(p > 1\) for which the sequence has a limit.

Consider the alternating series $$ \sum_{k=1}^{\infty}(-1)^{k+1} a_{k}, \text { where } a_{k}=\left\\{\begin{array}{cl} \frac{4}{k+1}, & \text { if } k \text { is odd } \\ \frac{2}{k}, & \text { if } k \text { is even } \end{array}\right. $$ a. Write out the first ten terms of the series, group them in pairs, and show that the even partial sums of the series form the (divergent) harmonic series. b. Show that \(\lim _{k \rightarrow \infty} a_{k}=0\) c. Explain why the series diverges even though the terms of the series approach zero.

Determine whether the following statements are true and give an explanation or counterexample. a. \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) is a convergent geometric series. b. If \(a\) is a real number and \(\sum_{k=12}^{\infty} a^{k}\) converges, then \(\sum_{k=1}^{\infty} a^{k}\) converges. If the series \(\sum_{k=1}^{\infty} a^{k}\) converges and \(|a|<|b|,\) then the series \(\sum_{k=1}^{\infty} b^{k}\) converges. d. Viewed as a function of \(r,\) the series \(1+r^{2}+r^{3}+\cdots\) takes on all values in the interval \(\left(\frac{1}{2}, \infty\right)\) e. Viewed as a function of \(r,\) the series \(\sum_{k=1}^{\infty} r^{k}\) takes on all values in the interval \(\left(-\frac{1}{2}, \infty\right)\)

After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of compensation is described by the relationship $$x_{n+1}=\frac{1}{2}\left(x_{n}+x_{n-1}\right), \quad \text { for } n=1,2,3, \ldots.$$ where \(x_{n}\) is the number of hours of sleep you get on the \(n\) th night and \(x_{0}=7\) and \(x_{1}=6\) are the number of hours of sleep on the first two nights, respectively. a. Write out the first six terms of the sequence \(\left\\{x_{n}\right\\}\) and confirm that the terms alternately increase and decrease. b. Show that the explicit formula $$x_{n}=\frac{19}{3}+\frac{2}{3}\left(-\frac{1}{2}\right)^{n}, \text { for } n \geq 0.$$ generates the terms of the sequence in part (a). c. What is the limit of the sequence?

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$
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