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 .

A (n) = 4 + (n - 1) 7

Substitute

$n = 53$

A (53) = 4 + (53 - 1) 7

▼

Evaluate right-hand side

SubTerm

Subtract term

A (53) = 4 + (52) 7

Multiply

A (53) = 4 + 364

AddTerms

Add terms

A (53) = 368

This means that the

36 8^{th}

day of the year will be the

5 3^{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 \cdot 7

Therefore, the year cannot have

53

Fridays unless the first day of that year is a Friday.

Exercise