Problem 43 - Algebra 2 and Trigonometry

Question

January \(1,2008,\) was a Tuesday. a. List the dates for each Tuesday in January of that year. b. Write a recursive definition for this sequence.

Step-by-Step Solution

Answer

The Tuesdays in January 2008 are January 1, 8, 15, 22, and 29. The recursive formula is \(a_1=1\) and \(a_{n+1}=a_n+7\).

Step 1: Identify the Pattern

The first step is to identify the pattern of Tuesdays in January 2008. Knowing that January 1, 2008, was a Tuesday, and days of the week follow a seven-day cycle, subsequent Tuesdays will occur every 7 days.

Step 2: Calculate Successive Tuesdays

Starting from January 1, 2008, add 7 days to find each subsequent Tuesday in the month: - January 1 + 7 = January 8 - January 8 + 7 = January 15 - January 15 + 7 = January 22 - January 22 + 7 = January 29 Thus, the Tuesdays in January 2008 are January 1, 8, 15, 22, and 29.

Step 3: Define Recursive Formula

To define a recursive formula for the sequence, let the sequence be represented as \(a_n\), where \(a_1 = 1\) (the first Tuesday, January 1). The recursive formula for this sequence will be \(a_{n+1} = a_n + 7\) for \(n \geq 1\). This formula correctly generates each subsequent Tuesday by adding 7 to the previous Tuesday's date.

Key Concepts

Recursive Formula

A recursive formula is a way to define the terms of a sequence using the preceding terms. When dealing with dates, such as the Tuesdays in January 2008, it's especially useful.
For example, the sequence of Tuesdays can be described recursively, which means each term in the sequence is related to the one before it.

• The first term, denoted as \(a_1\), is January 1st.
• Each subsequent Tuesday can be found by adding 7 days to the previous Tuesday. Mathematically, this is expressed as \(a_{n+1} = a_n + 7\), where \(a_n\) is the \(n^{th}\) Tuesday, and \(n \geq 1\).

This type of formula is especially useful because once you know the initial term and the rule for finding the next term, you can easily generate any term in the sequence. It's a step-by-step way of reaching a solution.

Calendar Calculations

Calendar calculations help us determine the days on which certain dates will fall. In recurring patterns like weekly cycles, they are very straightforward. Knowing the structure of the calendar—a seven-day week—you can easily determine the day of the week for any upcoming date given a starting point.

• For example, with January 1, 2008, known as a Tuesday, each subsequent Tuesday can be found by simply adding 7 days.
• This cycle reflects the basic rhythm of the calendar weeks which repeats every seven days.

By understanding this pattern, we can accurately predict future days and dates without consulting a calendar. This is particularly useful when planning events on specific days of the week.

Sequence Pattern

A sequence pattern in mathematics is simply an ordered list of numbers that follow a particular rule. In the context of our exercise, the sequence pattern refers to the set of dates that are Tuesdays in January 2008.

• The pattern here is simple but powerful: each date follows from the previous by adding a constant number, which is 7 in this case (because of the weekly cycle).
• So the sequence is 1, 8, 15, 22, and 29 of January. Each date is 7 days apart from the last.

Understanding sequence patterns allows us to predict future terms effortlessly. This regularity makes it a fundamental concept, not just in mathematics, but in various areas where recurring patterns occur. Identifying and using these patterns can simplify what initially seems complex. This is exactly what happens with our Tuesday dates for January 2008.