starting salary for a job is 30,000 with a guaranteed raise of $5,000 each year for the first 5 years. write an explicit formula to represent how much money you would earn each year. how much money would you earn in the fifth year?
To find the explicit formula for the amount of money earned each year, we need to consider the starting salary and the guaranteed raise.
The starting salary is $30,000. This means that in the first year, you would earn $30,000.
The guaranteed raise is $5,000 each year for the first 5 years. So, in the second year, you would earn the starting salary ($30,000) plus the raise ($5,000), which would give a total of $35,000. In the third year, you would earn $40,000 ($35,000 + $5,000). This pattern continues for the fourth and fifth years.
To represent this pattern as a formula, we can use a recursive equation:
M(n) = M(n-1) + $5,000
where M(n) represents the amount of money earned in year n and M(n-1) represents the amount of money earned in the previous year.
Now, let's apply the formula to find the amount of money earned in the fifth year:
M(5) = M(5-1) + $5,000
= M(4) + $5,000
To find M(4), we need to use the formula again:
M(4) = M(4-1) + $5,000
= M(3) + $5,000
Again, applying the formula:
M(3) = M(2) + $5,000
= M(1) + $5,000 + $5,000
= $30,000 + $5,000 + $5,000
Simplifying further:
M(3) = $40,000
Now we can substitute this value back into the earlier calculations to find M(4) and M(5):
M(4) = M(3) + $5,000
= $40,000 + $5,000
= $45,000
M(5) = M(4) + $5,000
= $45,000 + $5,000
= $50,000
So, in the fifth year, you would earn $50,000.