A new computer software company earns a profit of $245 000 in its first year. The company expects the profit to increase by 15% each year for each subsequent year.
(a) What profit can the company expect to earn in its seventh year?
(b) Find the total profit the company will earn in its first ten years.
in first year:
245,000
in second year
245,000 (1.15)^1
in third year
245,000 (1.15)^2
in nth year
245,000 (1.15)^(n-1)
NOW - go to web site Math is fun and look upo geometric sequence with
a = 245,000
and
r = 1.15
it also shows you how to sum the first ten terms :)
http://www.mathsisfun.com/algebra/sequences-sums-geometric.html
Thanks Damon!
To find the profit the company can expect to earn in its seventh year, we will use the formula for compound interest.
(a) The profit in the first year is $245,000. We need to find the profit in the seventh year, so we will calculate the compound interest over 6 years.
The formula for compound interest is: A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial profit)
r = the annual interest rate (15% = 0.15)
n = the number of times that interest is compounded per year (assuming it is compounded annually, so n = 1)
t = the number of years the money is invested
Using this formula, we can calculate the profit in the seventh year:
A = 245,000(1 + 0.15/1)^(1*6)
A = 245,000(1.15)^6
A ≈ $605,517.29
So, the company can expect to earn approximately $605,517.29 in its seventh year.
(b) To find the total profit the company will earn in its first ten years, we need to calculate the profit for each year and then sum them up.
The profits for each year can be calculated using the formula mentioned above. We will calculate and sum the profits for years 1 to 10:
Profit in year 1 = $245,000
Profit in year 2 = 245,000(1 + 0.15/1)^(1*1) ≈ $281,750.00
Profit in year 3 = 281,750(1 + 0.15/1)^(1*1) ≈ $323,012.50
Profit in year 4 = 323,012.50(1 + 0.15/1)^(1*1) ≈ $370,464.38
Profit in year 5 = 370,464.38(1 + 0.15/1)^(1*1) ≈ $424,034.02
Profit in year 6 = 424,034.02(1 + 0.15/1)^(1*1) ≈ $486,638.12
Profit in year 7 = 486,638.12(1 + 0.15/1)^(1*1) ≈ $559,634.84
Profit in year 8 = 559,634.84(1 + 0.15/1)^(1*1) ≈ $644,580.08
Profit in year 9 = 644,580.08(1 + 0.15/1)^(1*1) ≈ $743,311.09
Profit in year 10 = 743,311.09(1 + 0.15/1)^(1*1) ≈ $857,807.75
Total profit = 245,000 + 281,750.00 + 323,012.50 + 370,464.38 + 424,034.02 + 486,638.12 + 559,634.84 + 644,580.08 + 743,311.09 + 857,807.75
Total profit ≈ $5,956,233.68
So, the company will earn approximately $5,956,233.68 in its first ten years.
To determine the profit the company can expect to earn in its seventh year, we need to calculate the profit for each subsequent year, starting from year 1 and increasing by 15% each year.
(a) To find the profit in the seventh year, we can use the formula for compound interest, which is:
Profit = Principal * (1 + Rate)^Time
In this case:
- Principal (P) is the initial profit earned in the first year ($245,000)
- Rate (r) is the increase in profit each year (15% or 0.15)
- Time (t) is the number of years (7)
Using this formula, we can calculate the profit in the seventh year:
Profit = $245,000 * (1 + 0.15)^7
Calculating this expression, we find:
Profit = $245,000 * (1.15)^7
≈ $658,355.79
Therefore, the company can expect to earn a profit of approximately $658,355.79 in its seventh year.
(b) To find the total profit the company will earn in its first ten years, we need to sum up the profits for each year from year 1 to year 10.
To calculate the profit for each year, we can use the same formula as before:
Profit = Principal * (1 + Rate)^Time
In this case, we need to calculate the profit for years 1 to 10, so Time (t) will range from 1 to 10.
To find the total profit, we sum up all the individual year profits:
Total Profit = Profit in Year 1 + Profit in Year 2 + ... + Profit in Year 10
Using the formula, we can calculate the profit for each year and then sum them up to find the total.
Profit in Year 1 = $245,000
Profit in Year 2 = $245,000 * (1 + 0.15)^2
Profit in Year 3 = $245,000 * (1 + 0.15)^3
...
Profit in Year 10 = $245,000 * (1 + 0.15)^10
To find the total profit, we sum up the individual year profits:
Total Profit = $245,000 + $245,000 * (1 + 0.15)^2 + $245,000 * (1 + 0.15)^3 + ... + $245,000 * (1 + 0.15)^10
Evaluating this expression will give us the total profit the company will earn in its first ten years.