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?

The answer to the question is in the powers of 2.

21=2
22=4
23=8
24=16
.....
218=?
.....
2n=?

36 what is 18x2?

(a) To find out how much money you will receive on your 18th birthday, we need to calculate the pattern in which the amount doubles every year.

The amount of money received follows a geometric sequence, where each term is double the previous term. The first term is $2, and the common ratio is 2.

To find the 18th term, we can use the formula for the nth term of a geometric sequence:

Tn = a * r ^ (n-1)

Where:
Tn represents the nth term,
a represents the first term,
r represents the common ratio,
and n represents the term number.

So plugging in the values:
a = 2
r = 2
n = 18

T18 = 2 * (2) ^ (18-1)
T18 = 2 * 2 ^ 17
T18 = 2 * 131,072
T18 = 262,144

Therefore, you will receive $262,144 on your 18th birthday.

(b) To find out how much money you will receive on any given birthday, we can still use the formula for the nth term of a geometric sequence:

Tn = a * r ^ (n-1)

Where:
Tn represents the nth term,
a represents the first term,
r represents the common ratio,
and n represents the term number.

Once again, we have:
a = 2 (the first term)
r = 2 (the common ratio)

To find the amount of money you will receive on any given birthday (n), you simply plug in the value of n in the formula:

Tn = 2 * (2) ^ (n-1)

For example, to find out how much money you will receive on your 10th birthday, you would substitute n = 10 into the formula:

T10 = 2 * 2 ^ (10-1)
T10 = 2 * 2 ^ 9
T10 = 2 * 512
T10 = 1024

Therefore, you will receive $1024 on your 10th birthday.

(a) To find out how much money you will receive on your 18th birthday, we can observe a pattern in the amounts given on each birthday. Each amount is double the previous amount. So, you received $2 on your first birthday, which means:

2 * 2 = $4 on your second birthday
4 * 2 = $8 on your third birthday
8 * 2 = $16 on your fourth birthday

Using the same pattern, we can continue doubling the amounts until we reach the 18th birthday:

16 * 2 = $32 on your fifth birthday
32 * 2 = $64 on your sixth birthday
64 * 2 = $128 on your seventh birthday
128 * 2 = $256 on your eighth birthday
256 * 2 = $512 on your ninth birthday
512 * 2 = $1024 on your tenth birthday
1024 * 2 = $2048 on your eleventh birthday
2048 * 2 = $4096 on your twelfth birthday
4096 * 2 = $8192 on your thirteenth birthday
8192 * 2 = $16384 on your fourteenth birthday
16384 * 2 = $32768 on your fifteenth birthday
32768 * 2 = $65536 on your sixteenth birthday
65536 * 2 = $131072 on your seventeenth birthday
131072 * 2 = $262144 on your eighteenth birthday

So, on your 18th birthday, you will receive $262,144.

(b) To find out how much money you will receive on any given birthday (n), you can use the formula:

Amount on the nth birthday = 2^(n-1)

In this formula, "n" represents the birthday number. For example, n=1 for the first birthday, n=2 for the second birthday, and so on.

To use this equation, simply substitute the desired value of n into the formula and calculate the result. For instance, if you want to know how much money you will receive on your 10th birthday, you would substitute n=10 into the equation:

Amount on the 10th birthday = 2^(10-1) = 2^9 = 512

Therefore, you will receive $512 on your 10th birthday.