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.
Answers
MathMate
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 2<sup>n</sup>, and the total up to that day is 2<sup>n+1</sup>-2.
So on the hundredth day, the total would be 2<sup>101</sup>-2.
You can leave it in the above (exponent) form. If your teacher insists in getting a numeric answer, continue.
To calculate 2<sup>101</sup> - 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 2<sup>32</sup>, write down the number. Square again to get 2<sup>64</sup>. Multiply by 2<sup>32</sup>, which results in 2<sup>96</sup>. Multiply this by 2<sup>5</sup>=32 to get 2<sup>101</sup> 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
2<sup>101</sup>-2 = 2535301200456458802993406410750
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 2<sup>n</sup>, and the total up to that day is 2<sup>n+1</sup>-2.
So on the hundredth day, the total would be 2<sup>101</sup>-2.
You can leave it in the above (exponent) form. If your teacher insists in getting a numeric answer, continue.
To calculate 2<sup>101</sup> - 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 2<sup>32</sup>, write down the number. Square again to get 2<sup>64</sup>. Multiply by 2<sup>32</sup>, which results in 2<sup>96</sup>. Multiply this by 2<sup>5</sup>=32 to get 2<sup>101</sup> 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
2<sup>101</sup>-2 = 2535301200456458802993406410750
MathMate
For verification purposes,
2<sup>32</sup>=4294967296
2<sup>64</sup>=18446744073709551616, and
2<sup>101</sup>=2535301200456458802993406410752
2<sup>32</sup>=4294967296
2<sup>64</sup>=18446744073709551616, and
2<sup>101</sup>=2535301200456458802993406410752