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Chapter 8: Problem 28

Several terms of a sequence $$\left\\{a_{n}\right\\}_{n=1}^{\infty} \text { are given.}$$ a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. $$\\{1,4,9,16,25, \dots\\}$$

Short Answer

Expert verified
**Answer:** The next two terms of the sequence are 36 and 49. The recurrence relation is \(a_{n+1} = a_n + (2n + 1)\) for \(n \geq 1\), with \(a_1 = 1\). The explicit formula for the nth term of the sequence is \(a_n = n^2\).

Step by step solution

01

Identify the pattern and confirm it:

The given sequence is $$\\{1, 4, 9, 16, 25, ......\\}$$ Let's look for a pattern. It seems that each term in the sequence could be the square of a consecutive integer. Check if this is true: $$\\{1^2, 2^2, 3^2, 4^2, 5^2, ......\\} = \\{1, 4, 9, 16, 25, ......\\}$$ This matches the given sequence, so our suspicion was correct. a.
02

Find the next two terms of the sequence:

We now know that the sequence is formed by the squares of consecutive integers. So, we will find the next two terms by squaring the next two consecutive integers. For the term following 25 (\(5^2\)), we have: $$6^2 = 36$$ And, for the term following 36 (\(6^2\)), we have: $$7^2 = 49$$ So, the next two terms of the sequence are 36 and 49. b.
03

Find a recurrence relation for the sequence:

To find a recurrence relation, we can find the difference between consecutive terms. $$4 - 1 = 3$$ $$9 - 4 = 5$$ $$16 - 9 = 7$$ $$25 - 16 = 9$$ The differences between consecutive terms are odd numbers. Let's express this pattern as a recurrence relation. The term \(a_1 = 1\). For \(n \geq 1\), we can write the recurrence relation as: $$a_{n+1} = a_n + (2n + 1)$$ c.
04

Find an explicit formula for the nth term of the sequence:

Since we found that each term in the sequence is the square of a consecutive integer, we can write an explicit formula for the nth term of the sequence as follows: $$a_n = n^2$$

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Key Concepts

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

Recurrence Relation
A recurrence relation is an equation that expresses each term of a sequence in terms of its preceding terms. It acts as a formula that defines how each term builds on the previous one.

To find the recurrence relation of a sequence, we look at the difference or ratio between each consecutive term and determine a consistent pattern.

In the given sequence, \( \{1, 4, 9, 16, 25, \ldots \} \), the differences between the consecutive terms are odd integers: 3, 5, 7, and 9.

Knowing that, we can deduce the recurrence relation by noting that each term is derived from the sum of the previous term and an increasing odd number.

The recurrence relation for this sequence is:
  • The initial term: \( a_1 = 1 \)
  • For \( n \geq 1 \): \( a_{n+1} = a_n + (2n + 1) \)
By using this recurrence relation, each subsequent term is constructed by adding the next odd number to the current term.
Explicit Formula
An explicit formula is a direct expression that allows us to find the nth term of a sequence without having to know any of the previous terms. It gives us a straightforward way to calculate any term in the sequence by plugging in the term number directly.

In the original sequence \( \{1, 4, 9, 16, 25, \ldots\} \), we notice each term is the square of a positive integer in order.聽

This observation gives rise to the explicit formula for the nth term:
  • \( a_n = n^2 \)
Using this formula, you can easily calculate, for example, the 3rd term as \( 3^2 = 9 \) or the 6th term as \( 6^2 = 36 \) without referring back to other terms in the sequence.

This feature of the explicit formula makes it very handy for quickly identifying specific terms in a sequence.
Pattern Recognition
Pattern recognition in sequences involves identifying a recurring rule or formula that governs the placement of elements within the sequence. It's the foundational step in analyzing any sequence as it guides how we deduce both the recurrence relation and the explicit formula.

For the given sequence \( \{1, 4, 9, 16, 25, \ldots \}\), pattern recognition starts with observing each term's relationship with previous ones. Here, we notice the sequence might represent squares of consecutive integers based on the elements provided, which are \( 1^2, 2^2, 3^2, 4^2, 5^2, \ldots \).

Finding this pattern unlocks both the recurrence relation and the explicit formula:
  • The pattern shows terms as perfect squares, directing to the explicit formula \( a_n = n^2 \).
  • This further leads to understanding incremental differences for crafting the recurrence relation.
By using pattern recognition, one can not only predict subsequent terms but also formalize them through mathematical expressions, providing tools both for prediction and deeper understanding.

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Most popular questions from this chapter

Consider the number \(0.555555 \ldots,\) which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 .\) b. Consider the number \(0.54545454 \ldots\), which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots ., n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form for \(0 . \overline{n_{1}} n_{2} \cdots n_{p}\) d. Try the method of part (c) on the number \(0 . \overline{123456789}=0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

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?

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.

A tank is filled with 100 L of a \(40 \%\) alcohol solution (by volume). You repeatedly perform the following operation: Remove 2 L of the solution from the tank and replace them with 2 L of \(10 \%\) alcohol solution. a. Let \(C_{n}\) be the concentration of the solution in the tank after the \(n\) th replacement, where \(C_{0}=40 \% .\) Write the first five terms of the sequence \(\left\\{C_{n}\right\\}\). b. After how many replacements does the alcohol concentration reach \(15 \% ?\). c. Determine the limiting (steady-state) concentration of the solution that is approached after many replacements.

The expression $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+}}}}.$$ where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of $$a+\frac{b}{a+\frac{b}{a+\frac{b}{a+\frac{b}{a+}}}}$$ where \(a\) and \(b\) are positive real numbers.
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