Start by writing an explicit rule that represents the sequence of the days in the year that are Fridays.
No.
Practice makes perfect
We have been told that the first Friday of a new year is the fourth day of that year. We need to determine if the year will have 53 Fridays regardless of whether or not it is a leap year. To do so, we will follow a two-step plan.
Write an explicit rule that represents the sequence of the days in the year that are Fridays.
Use the rule to decide if the year will have 53 Fridays regardless of whether or not it is a leap year.
Writing an Explicit Rule
Since 1 week consists of 7 days and the first Friday of the year is the fourth day of that year, we can show the sequence as the following.
This is an arithmetic sequence because there is a common difference between consecutive terms. We can write its rule using the explicit formula for an arithmetic sequence.
A(n)= A(1)+(n-1) d
Here, n is the term number, A(1) is the first term, and d is the common difference. By substituting A(1)= 4 and d= 7, we can complete writing the rule that represents the sequence of the days in the year that are Fridays.
A(n)= 4+(n-1) 7
Using the Rule
Let's use the rule to check if the year will have 53 Fridays. For this purpose, we will substitute 53 for n.
This means that the 368^\text{th} day of the year will be the 53^\text{rd} Friday of the year. However, it is not possible for a year to have 368 days, even for a leap year. Therefore, the year will not have 53 Fridays.
Alternative Solution
Number of Full Weeks in a 365-Day Year
When we split a 365-day year into weeks, we get 1 day and 52 full weeks.
365= 1+ 52* 7
Therefore, the year cannot have 53 Fridays unless the first day of that year is a Friday.
