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Chapter 8: Problem 10

Let \(b_{0}, b_{1}, b_{2}, \ldots\) be defined by the formula \(b_{n}=4^{n}\), for all integers \(n \geq 0 .\) Show that this sequence satisfies the recurrence relation \(b_{k}=4 b_{k-1}\), for all integers \(k \geq 1\).

Short Answer

Expert verified
The sequence \(b_n = 4^n\) satisfies the recurrence relation \(b_k = 4b_{k-1}\) for all integers \(k \geq 1\), as shown by plugging in the formula for \(b_n\) into the recurrence relation and confirming that both sides of the equation are equal.

Step by step solution

01

Write down the given formula and recurrence relation

We are given the formula for the sequence: \[b_n = 4^n\] And the recurrence relation to prove: \[b_k = 4b_{k-1}\]
02

Plug in the formula for \(b_n\) into the recurrence relation

To show that the sequence satisfies the given recurrence relation, we need to substitute the formula for \(b_n\) into the relation. This means replacing \(b_k\) with \(4^k\) and \(b_{k-1}\) with \(4^{k-1}\): \[4^k = 4(4^{k-1})\]
03

Simplify the equation

Let's simplify the right-hand side of the equation: \[4^k = 4 \cdot 4^{k-1} = 4^1 \cdot 4^{k-1}\] Recall that, when multiplying bases with the same exponent, we can add the exponents: \[4^k = 4^{1+(k-1)}\]
04

Compare both sides of the equation

Now, let's check if both sides of the equation are equal: \[4^k = 4^{1+(k-1)}\] \[4^k = 4^{k}\] Both sides of the equation are indeed equal, confirming that the sequence satisfies the recurrence relation.
05

Conclusion

We have shown that the given sequence \(b_n = 4^n\) satisfies the recurrence relation \(b_k = 4b_{k-1}\) by plugging in the formula for \(b_n\) into the recurrence relation and confirming that both sides of the equation are equal for all integers \(k \geq 1\).

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Key Concepts

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

Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the original problem, the sequence \(b_n = 4^n\) is a geometric sequence where each term is multiplied by 4, which is the common ratio.
  • The general form of a geometric sequence is given by: \( a, ar, ar^2, ar^3, \ldots \), where \(a\) is the first term and \(r\) is the common ratio.
  • In the case of \(b_n = 4^n\), we have \(a = 1\) and \(r = 4\).
Understanding the properties of geometric sequences helps in solving various mathematical problems. For example, you can identify patterns, make predictions, and even model exponential growth or decay processes.
Inductive Proofs
Inductive proofs are a method of mathematical proof typically used to establish a statement for all natural numbers. The process involves two main steps. First, proving the base case, and second, the inductive step.
The base case involves verifying the statement for the initial value. For our exercise, this would be checking if the formula holds for the smallest value, commonly \( k = 1 \).
  • Base Case: Plug \( k = 1 \) into the relation \(b_k = 4b_{k-1}\) and check if it holds true.
  • Inductive Step: Assume it holds for \( k \) and then show it must hold for \( k+1 \). This typically involves using the assumption to prove the next step.
When successful, induction proves the statement true for all relevant integers. In our problem, the setup primarily relied on algebraic verification rather than a full formal induction due to the straightforward nature of powers of 4.
Integer Sequences
Integer sequences form the set of numbers where each member of the sequence is an integer. They are sometimes labeled with a formula to define the sequence, much like \(b_n = 4^n\) in the exercise.
Understanding integer sequences involves recognizing the pattern or formula that describes them:
  • Recursive Definition: Often, sequences are defined by a recurrence relation that specifies each term based on preceding terms, which is evident in our problem where \(b_k = 4b_{k-1}\).
  • Closed Form Definition: Provides a direct formula for the \(n\)-th term, such as \(b_n = 4^n\), allowing easy calculation without previous terms.
Integer sequences capture a wide array of numerical patterns and serve important roles in number theory, combinatorics, and many areas of mathematics. They provide a foundation for understanding more complex mathematical constructs.

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Most popular questions from this chapter

Counting Sirings: Consider the set of all strings of \(a^{\prime} \mathrm{s}, b^{\prime} \mathrm{s}\), and \(c^{\prime} s\). a. Make a list of all of these strings of lengths zero, one, two, and three that do not contain the pattern aa. b. For each integer \(n \geq 0\), let \(s_{n}=\) the number of strings of \(a^{\prime} s, b^{\prime} s\), and \(c^{\prime} s\) of length \(n\) that do not contain the pattern \(a a\). Find \(s_{0}, s_{1}, s_{2}\), and \(s_{3}\). \(H\) c. Find a recurrence relation for \(s_{0}, s_{1}, s_{2}, \ldots .\) d. Use the results of parts (b) and (c) to find the number of strings of \(a^{\prime} s, b^{\prime} s\), and \(c^{\prime} s\) of length four that do not contain the pattern aa.

A sequence is defined recursively. Use iteration to guess an explicit formula for the sequence. Use the formulas from Section \(4.2\) to simplify your answers whenever possible. $$ \begin{aligned} &x_{k}=3 x_{k-1}+k, \text { for all integers } k \geq 2 \\ &x_{1}=1 \end{aligned} $$

Find the first four terms of each of the recursively defined sequences in 1-8. \(b_{k}=b_{k-1}+3 k\), for all integers \(k \geq 2\) \(b_{1}=1\)

Suppose the population of a country increases at a steady rate of \(3 \%\) per year. If the population is 50 million at a certain time, what will it be 25 years later?

In each of 11-15 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. \(r_{k}=2 r_{k-1}-r_{k-2}\), for all integers \(k \geq 2\) \(r_{0}=1, r_{1}=4\)
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