your aunt gives you a check on your birthday each year. on your first birthday , she gives you a check for one cent. the next year she gives you 3 cents , then 9. each year she triples the amount of the previous year 's check. how old will you be when she gives you a check worth more than 1000$ ? how much money will you receive on your 18th birthday?

1c. = 3^0,

3c = 3^1,
9c. = 3^2,
27c = #^3.
So we have base 3 numerals.
A = 3^(n-1). A = Amount in cents @ nth year.

a. 3^(n-1) = 100,000 cents.
(n-1)Log3 = Log100,000,
n-1 = Log100,000 / Log3 = 10.5,
n = 11.5 years of age.

b. A = 3^(18-1),
A = 3^17 = 129,140,163 cents,
A = $1,291,401.63.

To determine how old you will be when your aunt gives you a check worth more than $1000, we need to find the progression of the amounts she gives you each year. Let's calculate:

First birthday: $0.01 (start)
Second birthday: $0.03 (triple the amount from the previous year)
Third birthday: $0.09 (triple the amount from the previous year)
Fourth birthday: $0.27 (triple the amount from the previous year)
Fifth birthday: $0.81
Sixth birthday: $2.43
Seventh birthday: $7.29
Eighth birthday: $21.87
Ninth birthday: $65.61
Tenth birthday: $196.83
Eleventh birthday: $590.49

By the eleventh birthday, your aunt's check would be $590.49. To find out when the amount exceeds $1000, we can continue the progression:

Twelfth birthday: $1,771.47
Thirteenth birthday: $5,314.41
Fourteenth birthday: $15,943.23
Fifteenth birthday: $47,829.69
Sixteenth birthday: $143,489.07
Seventeenth birthday: $430,467.21

On your seventeenth birthday, you will receive a check worth $430,467.21. This amount exceeds $1000.

Therefore, you will be 17 years old when your aunt gives you a check worth more than $1000.

Next, let's calculate how much money you will receive on your 18th birthday. To find the amount, we need to continue the progression:

Eighteenth birthday: $1,291,401.63

So, on your 18th birthday, you will receive a check worth $1,291,401.63.