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Chapter 8: Problem 3

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. \(a_{k}=k a_{k-1}\), for all integers \(k \geq 1\) \(a_{0}=1\)

Short Answer

Expert verified
The explicit formula for the given recursive sequence is \(a_{k} = k!\) for all integers \(k \geq 0\).

Step by step solution

01

Calculate the first few terms of the sequence

Start by finding the first few terms of the sequence using the recursive relation provided: \( a_{0} = 1 \) (Given) \( a_{1} = 1 * a_{0} = 1 \) \( a_{2} = 2 * a_{1} = 2 \) \( a_{3} = 3 * a_{2} = 3 * 2 = 6 \) \( a_{4} = 4 * a_{3} = 4 * 6 = 24 \) The first few terms of the sequence are 1, 1, 2, 6, 24,....
02

Identify the pattern

Now let's try to identify a pattern in the sequence obtained in the previous step: \( a_{0} = 1 \) \( a_{1} = 1 \) \( a_{2} = 1 * 2 \) \( a_{3} = 1 * 2 * 3 \) \( a_{4} = 1 * 2 * 3 * 4 \) We can observe that \( a_{k} \) is the product of all integers from 1 to k, which is the definition of the factorial of k, denoted by \( k! \).
03

Write the explicit formula

Based on the pattern identified, we can write the explicit formula for the given recursive sequence as: \( a_{k} = k! \) for all integers \( k \geq 0 \) Now, the explicit formula for the sequence is \( a_{k} = k! \). This formula is already simplified, and there is no need to simplify it further using the formulas from Section 4.2.

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Key Concepts

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

Factorial Function
The factorial function is a fundamental concept in mathematics that involves multiplying a sequence of descending positive integers. For a given positive integer \( n \), the factorial, denoted by \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Here's how the factorial function works:
  • It's defined for non-negative integers \( n \).
  • The value of \( 0! \) is defined as \( 1 \).
  • Factorials grow very quickly with larger values of \( n \).

The factorial function is widely used in permutations, combinations, and in solving problems where the arrangement of objects is considered. It also plays a major role in calculus and discrete mathematics.
Recursive Sequences
Recursive sequences are sequences where each term is defined using the preceding terms. The beauty of recursive sequences lies in their simplicity and power. Specifically, they can economize on computation by using previously computed values.

In the given sequence, \( a_k = k \times a_{k-1} \), each term is formed by multiplying the previous term by \( k \). This reflects a recursive approach where:
  • A base case or starting point is essential. For this sequence, \( a_0 = 1 \) serves as the initial condition.
  • Each subsequent value is computed using the preceding values step by step.

This recursive definition mirrors processes found in numerous natural phenomena, making it an integral part of computer algorithms, mathematical modeling, and various simulations.
Explicit Formula Derivation
Deriving an explicit formula from a recursive one provides a direct way to calculate terms without iterating through all previous terms. For our sequence, the recursive formula \( a_k = k \times a_{k-1} \) can be translated into an explicit formula.

Through experimentation with initial terms:
  • \( a_0 = 1 \)
  • \( a_1 = 1 \times 1 = 1 \)
  • \( a_2 = 2 \times 1 = 2 \)
  • \( a_3 = 3 \times 2 = 6 \)
  • \( a_4 = 4 \times 6 = 24 \)

Pattern recognition reveals that each term corresponds to the factorial of its index, leading to the explicit formula \(a_k = k!\). This clear and simplified formula allows us to compute any term in sequence without referring to its predecessors, underpinning the utility of deriving explicit formulas from recursive definitions.

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Most popular questions from this chapter

Refer to the sequence of Stirling numbers of the second kind. Find the total number of different partitions of a set with five elements.

Four-Pole Tower of Hanoi: Suppose that the Tower of Hanoi problem has four poles in a row instead of three. Disks can be transferred one by one from one pole to any other pole, but at no time may a larger disk be placed on top of a smaller disk. Let \(s_{n}\) be the minimum number of moves needed to transfer the entire tower of \(n\) disks from the left-most to the right-most pole. a. Find \(s_{1}, s_{2}\), and \(s_{3}\). b. Find \(s_{4}\) c. Show that \(s_{k} \leq 2 s_{k-2}+3\) for all integers \(k \geq 3\).

A person borrows \(\$ 3,000\) on a bank credit card at a nominal rate of \(18 \%\) per year, which is actually charged at a rate of \(1.5 \%\) per month. a. What is the annual percentage rate (APR) for the card? (See Example 8.1.8 for a definition of APR.) b. Assume that the person does not place any additional charges on the card and pays the bank \(\$ 150\) each month to pay off the loan. Let \(B_{n}\) be the balance owed on the card after \(n\) months. Find an explicit formula for \(B_{n}\). c. How long will be required to pay off the debt? d. What is the total amount of money the person will have paid for the loan?

Define a set \(S\) recursively as follows: I. BASE: \(\in \in S\) II. RECURSION: If \(s \in S\), then a. \(b s \in S\) b. \(s b \in S\) c. \(s a a \in S\) d. aas \(\in S\) III. RESTRICTION: Nothing is in \(S\) other than objects defined in I and II above. Use structural induction to prove that every string in \(S\) contains an even number of \(a\) 's.

A sequence is defined recursively. (a) Use iteration to guess an explicit formula for the sequence. (b) Use strong mathematical induction to verify that the formula of part (a) is correct. \(b_{k}=\frac{2}{b_{k-1}}\), for all integers \(k \geq 2\) $$ b_{1}=1 $$
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