Problem 80 - Algebra for College Students

Question

$$ \left\\{\begin{array}{l} a_{1}=3 \\ a_{n}=a_{n-1}+2 \quad \text { for } n \geq 2 \end{array} 5,7,9,11,13\right. $$

Step-by-Step Solution

Answer

The sequence is correct: it starts with 3, and terms are increased by 2, matching the given numbers after the first term.

Step 1: Understanding the Sequence Definition

The sequence is defined recursively with the first term \(a_1 = 3\) and a relation for subsequent terms \(a_n = a_{n-1} + 2\), meaning each term is 2 more than the previous term.

Step 2: Finding the First Few Terms

Based on the sequence definition, let's calculate the first few terms:- \(a_1 = 3\).- \(a_2 = a_1 + 2 = 3 + 2 = 5\).- \(a_3 = a_2 + 2 = 5 + 2 = 7\).- \(a_4 = a_3 + 2 = 7 + 2 = 9\).- \(a_5 = a_4 + 2 = 9 + 2 = 11\).- \(a_6 = a_5 + 2 = 11 + 2 = 13\).

Step 3: Confirming the Sequence Matches the Given Terms

Compare the sequence calculated to the provided numbers \(5, 7, 9, 11, 13\). The terms in the sequence starting from \(a_2\) are \(5, 7, 9, 11, 13\), which match perfectly.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference of any two successive numbers is a constant. This difference is known as the common difference. In the given exercise, the common difference is 2.

• Each term increases by a set amount: this is the defining feature of an arithmetic sequence.
• This consistent increase makes it easy to predict future terms.

Knowing the first term and the common difference allows for easy generation of the entire sequence by simply adding the difference repeatedly to the first term.

Sequence Definition

A sequence is simply a list of numbers in a specific order. Definition of a sequence can be either explicit or recursive. In this exercise, the sequence is defined recursively.

• The starting point is given: here, the first term is 3.
• This term serves as the base for building the remainder of the sequence.

With the first term as a foundation, subsequent terms are described through rules or formulas that precisely dictate how to find these terms.

Recursive Formula

A recursive formula expresses each term based on the preceding terms. This makes it distinct from an explicit formula, which defines each term individually.

• In our exercise, the recursive formula is: \(a_n = a_{n-1} + 2\).
• This formula tells us exactly how to construct each new term from the last.

Recursive formulas are especially useful for constructing sequences element by element, which can be helpful in solving iterative problems or computational algorithms.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. Although it is not explicitly mentioned in the exercise, understanding this concept can solidify your grasp on sequences.

• Induction typically involves two steps: the base case and the induction step.
• First, you prove the base case (often the first term of the sequence) is true.
• Next, you assume that a term \(k\) is true to show \(k+1\) is also true.

Using mathematical induction can validate the correctness of the sequence or pattern indefinitely, making it an invaluable tool in proving sequences work as expected in mathematics.