How many gifts will I have after twelve days of Christmas following the pattern of gifts given?

Understand the Problem

The question is asking us to calculate the total number of gifts received after twelve days, considering the pattern of receiving new gifts and copies of all previous gifts each day.

Answer

The total number of gifts received after twelve days is $4095$.

Steps to Solve

Identify the Gift Pattern
Each day, you receive one new gift plus a copy of all gifts received on previous days.
Calculate Gifts for Each Day
On Day 1, you receive 1 gift.
On Day 2, you receive 1 new gift + 1 copy of Day 1 gift = 2 gifts.
On Day 3, you receive 1 new gift + 1 copy of Day 1 gift + 1 copy of Day 2 gift = 4 gifts.

Continuing this pattern:

Day 1: 1 gift
Day 2: 2 gifts (Total: 1 + 2 = 3 gifts)
Day 3: 4 gifts (Total: 3 + 4 = 7 gifts)
Day 4: 8 gifts (Total: 7 + 8 = 15 gifts)
Day 5: 16 gifts (Total: 15 + 16 = 31 gifts)
Day 6: 32 gifts (Total: 31 + 32 = 63 gifts)
Day 7: 64 gifts (Total: 63 + 64 = 127 gifts)
Day 8: 128 gifts (Total: 127 + 128 = 255 gifts)
Day 9: 256 gifts (Total: 255 + 256 = 511 gifts)
Day 10: 512 gifts (Total: 511 + 512 = 1023 gifts)
Day 11: 1024 gifts (Total: 1023 + 1024 = 2047 gifts)
Day 12: 2048 gifts (Total: 2047 + 2048 = 4095 gifts)

Final Calculation
After twelve days, add the total gifts received each day:
$$ \text{Total gifts} = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 = 4095 $$

The total number of gifts received after twelve days is $4095$.

More Information

This problem illustrates exponential growth, as each day's gifts double the total received the previous day. The pattern is similar to the powers of 2, where the number of gifts received on each day can be represented as $2^{(n-1)}$ where $n$ is the day number.

Tips

One common mistake is forgetting to add all previous gifts correctly. It's important to maintain the doubling pattern without skipping any steps.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!