Problem 76 - Algebra for College Students

Question

$$ a_{n}= \begin{cases}\frac{1}{n} & \text { for } n \text { odd } \\ n^{2} & \text { for } n \text { cven } \\ 1,4, \frac{1}{3}, 16, \frac{1}{5}, 36\end{cases} $$

Step-by-Step Solution

Answer

The sequence alternates between \( \frac{1}{n} \) for odd \( n \) and \( n^2 \) for even \( n \).

Step 1: Identifying Odd and Even Numbers

First, identify whether each number in the sequence corresponds to an odd or even value of \( n \). The sequence given is \( 1, 4, \frac{1}{3}, 16, \frac{1}{5}, 36 \). Here, 1, \( \frac{1}{3} \), and \( \frac{1}{5} \) should align with odd positions, and 4, 16, and 36 should align with even positions.

Step 2: Mapping Sequence to Formula

Match each element to its corresponding formula based on the position:- **\( a_1 = \frac{1}{n} \) with \( n = 1 \) (odd), thus \( a_1 = 1 \).**- **\( a_2 = n^2 \) with \( n = 2 \) (even), thus \( a_2 = 4 \).**- **\( a_3 = \frac{1}{n} \) with \( n = 3 \) (odd), thus \( a_3 = \frac{1}{3} \).**- **\( a_4 = n^2 \) with \( n = 4 \) (even), thus \( a_4 = 16 \).**- **\( a_5 = \frac{1}{n} \) with \( n = 5 \) (odd), thus \( a_5 = \frac{1}{5} \).**- **\( a_6 = n^2 \) with \( n = 6 \) (even), thus \( a_6 = 36 \).**

Key Concepts

Odd and Even Numbers

Odd and even numbers are an essential concept in algebra and mathematics as a whole. They are used to classify numbers based on their divisibility by 2.

• An even number is a number that is divisible by 2 without leaving a remainder. Examples of even numbers include 2, 4, 6, etc.
• An odd number, on the other hand, is a number that leaves a remainder of 1 when divided by 2. Examples of odd numbers include 1, 3, 5, etc.

Understanding whether a number is odd or even is crucial when working with mathematical sequences or functions, especially when the behavior of a sequence changes based on the parity of its index. In recursive formulas or sequences, each position in the sequence might be defined differently based on whether the position index (denoted as \( n \)) is odd or even. Making this distinction helps to correctly determine the formula or rule to apply to each term in the sequence, as seen in the example problem.

Recursive Formulas

A recursive formula establishes how each term in a sequence is related to the previous terms. This connection allows us to find any term of the sequence if we know the preceding ones, or sometimes just the initial conditions. Recursive formulas are often used in sequences where relationships between terms are clearly defined. For instance, in our example problem, different expressions are used when the index \( n \) is either odd or even. This classification leads to two distinct types of formulas:

• For odd \( n \): The formula is given by \( a_n = \frac{1}{n} \).
• For even \( n \): The formula becomes \( a_n = n^2 \).

Each of these formulas defines how terms are calculated depending on the parity of \( n \). Recursive formulas can sometimes reference previous terms (like the Fibonacci sequence), but in this example, they primarily depend on the current index. Understanding how to apply recursive formulas can aid in determining the value of each element in a mathematical sequence efficiently.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers defined by a specific pattern or rule. They are fundamental in mathematics and are used to model a wide array of phenomena in different fields.In sequences, each number in the list is called a term. The position of the term in the sequence is referred to as its index, usually denoted as \( n \). When dealing with sequences, one of the primary tasks is to determine the nth term, based on some described formula or rule, just like the example problem. The example shows a mixed sequence that uses one rule for odd-indexed terms and another for even-indexed terms:

• For odd indices (1, 3, 5,...): Terms are calculated using the formula \( a_n = \frac{1}{n} \).
• For even indices (2, 4, 6,...): The formula \( a_n = n^2 \) is used.

Different types of sequences follow different rules or expressions, ranging from arithmetic and geometric sequences, where differences or ratios between successive terms are consistent, to more complex patterns defined recursively. Understanding sequences and their rules is crucial for algebra, making sense of progression, and identifying patterns and relationships between numbers.