Question

Mai and Andre found an old, brass bottle that contained a

magical genie. They freed the genie, and it offered them
each a magical $1 coin as thanks.
. The magic coin turned into 2 coins on the first day.
. The 2 coins turned into 4 coins on the second day.
. The 4 coins turned into 8 coins on the third
This doubling pattern continued for 28 days.
Mai was trying to calculate how many coins she would have and remembered that instead
of writing 1.2.2.2.2.2.2 for the number of coins on the 6th day, she could just
write 26.

Answer 1

Mai is correct. We can use exponential notation to represent the number of coins each day. Let's call the number of coins on the first day "1". Then the number of coins on each subsequent day is twice the number of coins on the previous day. So we have:

Day 1: 1

Day 2: 2 = 2^1

Day 3: 4 = 2^2

Day 4: 8 = 2^3

...

Day n: 2^(n-1)

To find the number of coins Mai has on the 6th day, we substitute n = 6 into the formula for the number of coins:

Day 6: 2^(6-1) = 2^5 = 32

So Mai has 32 coins on the 6th day. Writing out the product of 2's (1.2.2.2.2.2.2) is equivalent to writing 2^6 = 32.

To find out how many coins Mai has after 28 days, we substitute n = 28 into the formula for the number of coins:

Day 28: 2^(28-1) = 2^27 = 134,217,728

So after 28 days, Mai has 134,217,728 coins.