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 that represents the 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 because there is a 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.
A(n)=4+(n−1)7
A(53)=4+(53−1)7
A(53)=368
This means that the
368th day of the year will be the
53rd 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.
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.