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.

Answer 1

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!!!