Answer:
The right analytical model for the profits of the company for any given year n, (pₙ) is that of a geometric series and it is given as
pₙ = 3000 (3ⁿ⁻¹)
Step-by-step explanation:
Concluding part of the question
In year one, a company earned $3,000 in profits; in year two they earned $9,000 in profits. In year 3 they earned $27,000 in profits and in year 4 they earned $81,000. Which analytical model illustrates this profit pattern?
Solution
From the pattern of the profits, it is evident that the profits follow a geometric series' pattern with the first term equal to the profits in the first year, $3000, the profit in the second year is the second term and so on.
First term = a = Profits in the first year = 3000
Second term = a₂ = Profits in the second year = 9000
Third term = a₃ = Profits in the third year = 27000
Fourth term = a₄ = Profits in the fourth year = 81000
The general formula for any term of a geometric series is
aₙ = arⁿ⁻¹
a = first term
r = common ratio
But for a geometric series, common ratio is given as the next term divided by the very previous term.
r = (aₙ/aₙ₋₁)
For this question,
r = (9000/3000) = (27000/9000) = (81000/27000) = 3
So, the right model for the profits for any given year n is
pₙ = 3000 × 3ⁿ⁻¹
pₙ = 3000 (3ⁿ⁻¹)
Hope this Helps!!!
