Problem 75 - Contemporary Precalculus

Question

Deal with prime numbers. A positive integer greater than 1 is prime if its only positive integer factors are itself and 1. For example, 7 is prime because its only factors are 7 and \(1,\) but 15 is not prime because it has factors other than 15 and 1 (namely, 3 and 5 ). Find the first five terms of the sequence. \(a_{n}\) is the largest prime integer less than \(5 n\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 3, 7, 13, 19, and 23.

Step 1: Find the first five values of \(5n\).

Given the sequence defined by \(a_n\), calculate the first five values of \(5n\), corresponding to the term's index. For n = 1: \(5(1) = 5\) For n = 2: \(5(2) = 10\) For n = 3: \(5(3) = 15\) For n = 4: \(5(4) = 20\) For n = 5: \(5(5) = 25\)

Step 2: Find the prime numbers.

We'll now find the largest prime numbers less than each of the values obtained in Step 1. For 5: The prime numbers less than 5 are 2 and 3. The largest prime number among these is 3. For 10: The prime numbers less than 10 are 2, 3, 5, and 7. The largest prime number among these is 7. For 15: The prime numbers less than 15 are 2, 3, 5, 7, 11, and 13. The largest prime number among these is 13. For 20: The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. The largest prime number among these is 19. For 25: The prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19, and 23. The largest prime number among these is 23.

Step 3: List the first five terms.

The first five terms of the sequence (\(a_n\)) are: \(a_1 = 3\) \(a_2 = 7\) \(a_3 = 13\) \(a_4 = 19\) \(a_5 = 23\)

Key Concepts

Integer Sequences

An integer sequence is a list of integers that follow a specific pattern or rule. In this context, the sequence is defined by a function \(a_n\), where \(a_n\) represents the largest prime number less than \(5n\). Each term is calculated from the given rule relating to the index \(n\) of the sequence.
To find a term in this sequence:

• First, you calculate \(5n\). This gives you a reference number (like 5, 10, 15 in the step-by-step solution).
• Then, find all prime numbers less than this computed number.
• The last step is to choose the largest prime under the given number.

Understanding integer sequences like this one is key to recognizing patterns in numbers, and these patterns form the basis for many mathematical theories.

Factorization

Factorization is a process of breaking down numbers into their prime components. It involves expressing a number as a product of its prime factors. Prime numbers are integers greater than 1 that are only divisible by 1 and themselves.
For example:

• 15 can be factored into 3 and 5, which are both prime numbers, making 15 a composite number (non-prime).
• In each sequence term in the exercise, the numbers before finding the largest prime, like 10 or 20, are analyzed by their division into primes.

Factorization helps in the simplification of mathematical problems, identification of prime numbers, and is useful in solving algebraic equations. Recognizing that prime numbers appear less frequently in large numbers is essential in understanding factorization's role.

Algebraic Concepts

Algebraic concepts often involve manipulating mathematical expressions to find unknown values. In the exercise provided, we deal with an equation of a form like \(a_n = \text{largest prime less than } 5n\). This equation illustrates an algebraic rule that defines the sequence.
Steps to employ algebraic thinking include:

• Identifying the pattern or rule governing the sequence.
• Using the rule to compute numerical values, like \(5n\), and then perform additional operations (finding prime).
• Expressing the result conforming to algebraic equations or functions.

Algebra allows us to succinctly represent complex problems with symbols and equations. It also provides methodologies to systematically solve various mathematical puzzles, including sequences and factorization tasks.

Number Theory

Number theory is a branch of mathematics focusing on properties and relationships of numbers, particularly integers. The concept of prime numbers is central to number theory, as primes are the building blocks of natural numbers.
The problem presented about finding the largest prime less than 5n draws from various number theory concepts:

• Understanding what constitutes a prime number.
• Using prime numbers to break down and understand the nature of integer sequences.
• Exploring the density and distribution of primes, key subjects within number theory.

Number theory not only underpins structures like sequences but also extends into cryptography, coding theory, and more. Recognizing patterns, as seen in finding primes under specific constraints, showcases the beautiful complexity of number structures in mathematics.