**Carla’s Clothing Shop opened 8 years ago. The first year she made $3,000 profit. Each year thereafter her profits averaged 50% greeater than the previous year. 1. How much profits did Carla earn during her 18th year of business. 2. What was the total amount of profit Carla earned over her first 18 years?**

**Answer:**

Hello! This problem is a tad tricky to wrap one’s head around. But lets take a look.

So far, we know that Carla made a profit of $3000 during her first year. Knowing that, we know that the second year will have a profit of 50% greater than what she had during first year. This is stated by the question.

So it is fair to say that Carla’s profit during her second year will be:

Profit of Second Year = First Year Profit + 0.5 (First Year Profit)

= 1.5 (First Year Profit) , subbing First Year Profit

= 1.5 ( 3000)

= $4500.00

So Carla’s Profit during her second year is $4500.00.

What about her third year? So we doing the same thing again,

Profit of third year = 1.5 (Second Year Profit)

= 1.5 ( 4500)

= 6750

So during third year she made $6750.00.

What about her fourth year?

Fourth year profit = 1.5 ( Third year profit)

= 1.5 ( 6750)

= $10 125

So during her fourth year she made $10 125.00

Hmm a pattern seems to be occurring. For every year we just simply multiply 1.5 to the year prior! So lets create a formula that will determine the profit for all the years to come!

So let f(x) be the profit and x be the year and x cannot be smaller than 1

f(x) = (1.5 ^(x-1) ) * 3000

hmm this can be derived by seeing the pattern more precisely.

Second Year Profit = 1.5 * 3000

Third Year Profit = 1.5 * Second Year Profit

= 1.5 * 1.5 * 3000

Fourth Year Profit = 1.5 * 1.5 *1.5 * 3000

See the repetition of 1.5s?

Now how much did Carla earn during her 18th year?

lets use our function

f(18) = (1.5^(18-1)) * 3000

= (1.5^17) * 3000

= 2955783.76007

Since this is a real life based problem, there are no 7/10000 cent so we round down.

So during her 18th year, she made a profit of $2955783.76.

So for part 2.

the total amount of profit can be determined by summing f(1) + f(2) + … + f(18).