You take a job with a starting salary of $37,000. Your employer offers you an annual raise of $1500 for the next 6 years. Write a rule for your salary in the Nth year. What are your total earnings in 6 years?
How would you go about solving this problem? We're working on summation formulas, arithmetic and geometric sequences, and Sums of those if that helps!
x=37000+1500(n-1)
=37000+1500n -1500
x=1500n - 35500
first year: 37000
second year: 38500
third year: 40000
fourth year: 41500
fifth year: 43000
sixth year: 44500
these 6 years numbers are add together and it becomes: $244500
x=1500n-35500
i forgot to add that if you sub in 1-6 year in the n and you will get those answer above
Well, let me put on my clown hat and give you a funny answer to your question about salary and raises.
Why did the salary cross the road?
Because it wanted to escape the annual raise! 🤡
But jokes aside, let's get serious and solve this math problem.
To write a rule for your salary in the Nth year, we can use the arithmetic sequence formula:
Salary(N) = Starting Salary + (N - 1) * Annual Raise
In this case, the starting salary is $37,000 and the annual raise is $1500. So the rule for your salary in the Nth year would be:
Salary(N) = $37,000 + ($1500 * (N - 1))
To calculate your total earnings in 6 years, we need to find the sum of salaries from the first year to the sixth year. We can use the sum of an arithmetic sequence formula:
Sum = (n/2) * (2a + (n - 1)d)
Where n is the number of terms (in this case, 6), a is the first term (starting salary), and d is the common difference (annual raise). Plugging in our values, we get:
Sum = (6/2) * (2 * $37,000 + (6 - 1) * $1500)
Simplifying this equation gives us:
Sum = 3 * ($74,000 + 5 * $1500)
Sum = 3 * ($74,000 + $7,500)
Sum = 3 * $81,500
Sum = $244,500
So, your total earnings in 6 years would be $244,500.
Hope that helps, and remember, clowning around with math can make it more entertaining! 🤡
To find the rule for your salary in the Nth year, we can use an arithmetic sequence formula. In this scenario, the first term (a₁) is $37,000, and the common difference (d) is $1500. The formula for an arithmetic sequence is:
aₙ = a₁ + (n - 1) * d
where aₙ represents the salary in the Nth year.
Using the given values, we have:
aₙ = 37,000 + (n - 1) * 1500
To find the total earnings in 6 years, we need to calculate the sum of the arithmetic sequence from the 1st term (a₁) to the 6th term (a₆). The formula for the sum of an arithmetic sequence is:
Sn = (n/2) * (2a₁ + (n - 1) * d)
Substituting in the formula for aₙ, we get:
Sn = (n/2) * (2 * a₁ + (n - 1) * d)
Plugging in the values:
S₆ = (6/2) * (2 * 37,000 + (6 - 1) * 1500)
Now we can solve for the total earnings (S₆).
To solve this problem, we can start by writing a rule for the salary in the Nth year. From the given information, we know that the starting salary is $37,000, and the annual raise is $1500.
The rule for the salary in the Nth year can be written using an arithmetic sequence. The formula for an arithmetic sequence is:
an = a1 + (n - 1)d
where a1 is the first term, n is the number of terms (years in this case), d is the common difference (annual raise).
In this case, the first term (a1) is $37,000, and the common difference (d) is $1500. Therefore, the rule for the salary in the Nth year is:
Sn = 37000 + (N - 1) * 1500
Now, to find the total earnings in 6 years, we need to calculate the sum of the salaries for each year. We can use the formula for the sum of an arithmetic series:
Sn = (n / 2) * (a1 + an)
where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.
In this case, the number of terms (n) is 6, the first term (a1) is $37,000, and the last term (an) can be calculated using the rule for the salary in the Nth year by substituting N with 6:
an = 37000 + (6 - 1) * 1500 = 37000 + 5 * 1500 = 44500
Substituting the values into the formula, we get:
S6 = (6 / 2) * (37000 + 44500) = 3 * 81500 = $244,500
Therefore, your total earnings in 6 years would be $244,500.