Your great uncle gave you $2 on your first birthday, $4 on your 2nd birthday, $8 on your 3rd birthday, and so on.
(a) How much money will you receive on your 18th birthday?
(b) How much money will you receive on your nth birthday and explain how to use your equation to find how much money you will receive on any given birthday?
2^n
$170,880
(a) To find out how much money you will receive on your 18th birthday, we can use the formula for a geometric series:
S = a * (r^n - 1) / (r - 1)
Where:
S = the sum of the geometric series
a = the first term of the series
r = the common ratio
n = the number of terms in the series
In this case, the first term (a) is $2 and the common ratio (r) is 2 because each subsequent amount is double the previous amount.
Plugging in these values into the formula:
S = 2 * (2^18 - 1) / (2 - 1)
S = 2 * (262,143 - 1) / 1
S = 2 * 262,142
S = $524,284
Therefore, you will receive $524,284 on your 18th birthday.
(b) To find out how much money you will receive on your nth birthday, we can use the same formula and substitute 'n' for the number of terms.
S = a * (r^n - 1) / (r - 1)
To use this equation, you would substitute the values of 'a' and 'r' from the pattern given, and replace 'n' with the specific birthday number you want to calculate. The resulting value of 'S' will give you the amount of money you will receive on that specific birthday.
To find the amount of money you will receive on your 18th birthday, we can see that each year the amount doubles. So, on your 18th birthday, you will receive 2^17 dollars.
To find the amount of money you will receive on your nth birthday, we can use the formula: amount = 2^(n-1). Here, n represents the birthday number.
For example, if you want to find the amount of money you will receive on your 10th birthday, you would substitute n=10 in the formula: amount = 2^(10-1) = 2^9.
To calculate it, you can start with the initial amount of 2, and then keep multiplying by 2 for each additional year.
So, on your 10th birthday, you would calculate:
amount = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 512 dollars.