Question

a farmer is rewarded for saving the king. he can either get 1000 gold bars up front or payment over time for 100 days which begins at 2 gold bars the first day, 4 the second, 8 the third, 16 the fourth, and so on. which one should he choose and why. Show how you got the answer? i know that the second choice is best i just don't know how to do the math to see how much it will be in 100 days.

Answer 1

First, you try it for 5 days,
1 day = 2, total = 2
2 day = 2.2 = 4, total = 6
3 = 2.2.2 = 8, total = 14
4 = 2.2.2.2 = 16, total = 30
5 = 2.2.2.2.2 = 32, total = 62
You will note that the total is the same as the next-day's amount less two, i.e. to get the total of each day, you need to double the day's amount and subtract two.

So far so good. We have just concluded that the the amount for the nth day is 2ⁿ, and the total up to that day is 2ⁿ⁺¹-2.

So on the hundredth day, the total would be 2¹⁰¹-2.

You can leave it in the above (exponent) form. If your teacher insists in getting a numeric answer, continue.

To calculate 2¹⁰¹ - 2, you have many options,
1. Starting with 1, multiply by two 101 times (and subtract 2).
2. A shorter way is to start with 1, <i>square</i> 5 times to get 2³², write down the number. Square again to get 2⁶⁴. Multiply by 2³², which results in 2⁹⁶. Multiply this by 2⁵=32 to get 2¹⁰¹ and don't forget to subtract 2.
3. The third and simplest way is to use your calculator, if one that has 30 digits of accuracy is available.

The answer is
2¹⁰¹-2 = 2535301200456458802993406410750

Answer 2

For verification purposes,
2³²=4294967296
2⁶⁴=18446744073709551616, and
2¹⁰¹=2535301200456458802993406410752