In order to cover upfront expenses, Pinecrest Enterprises must be able to come up with $50 million of its own capital reserves. Presently, Pinecrest has $37 million in capital reserves available, and can add $1.5 million per month .
After how many months will Pinecrest Enterprises meet the reserve requirement? Round up to the next month.
OOPS
8
HINT
Find the formula for the amount the company has after n months. Hint: this can be expressed using an arithmetic sequence. Then, solve the appropriate equation when using this formula. Note that since the payments are made monthly, your answer must be an integer value.
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To determine how many months it will take for Pinecrest Enterprises to meet the reserve requirement, we can use the formula for an arithmetic sequence. The formula for the nth term of an arithmetic sequence is:
An = A1 + (n - 1) * d
Where An represents the nth term, A1 is the initial term, n is the number of terms, and d is the common difference between consecutive terms.
In this case, the initial term (A1) is $37 million and the common difference (d) is $1.5 million per month. Let's use this information to solve the equation for An. We want to find the smallest value of n for which An is equal to or greater than $50 million.
50 = 37 + (n - 1) * 1.5
Simplifying the equation:
12.5 = (n - 1) * 1.5
Dividing both sides of the equation by 1.5:
n - 1 = 8.33333...
To round up to the next month, we round n up to 9.
Therefore, it will take Pinecrest Enterprises 9 months to meet the reserve requirement of $50 million.