Expert: Sherman D. - 9/15/2004

Question
One day you went to visit Uncle Harry's farm. You went out to the outer most apple orchard and picked a basket of apples. To get back to the house you had to pass through 7 gates. At each gate there was a guard and in order to pass through you had to give him half of your apples and then one more. In other words, at the first gate you gave the guard half your apples and then one more. At the next gate you did the same thing until by the time you passed through the last gate you only had one apple left. *How many apples did you originally have in your basket?
ALSO I NEED TO KNOW:
1. What happens to the answer when you change the number of apples you have left at the end?
2. What happens to the answer when you change the number of gates you have to go through?
3. How many apples did you originally have in your basket if you had n apples left at the end and you went through m gates?
please and thank you!!!


Answer
n -
((1/2)n + 1) -
((1/4)n + (3/2)) -
((1/8)n + (7/4)) -
((1/16)n + (15/8)) -
((1/32)n + (31/16)) -
((1/64)n + (63/32)) -
((1/128)n + (96/64))
+ 1 =
0

If you are wondering where i got the values, just say half of the value before it, the add one, for ex:

(1/2)((1/2)n + 1) = (1/4)n + (1/2)
Now add 1, and you get it to shorten to
(1/4)n + (3/2)

Now say half of that

(1/2)((1/4)n + (3/2)) + 1
(1/8)n + (3/4) + 1
(1/8)n + (3/4) + (4/4)
(1/8)n + (7/4)

And continue until you get to the 7th gate

1.) Changing the apples at the end would change the number of apples you started out with.
2.) Changing the number of gates would change how many apples you start out with.

3.)
n -
((1/2)n + 1) -
((1/4)n + (3/2)) -
((1/8)n + (7/4)) -
((1/16)n + (15/8)) -
((1/32)n + (31/16)) -
((1/64)n + (63/32)) -
((1/128)n + (96/64)) + 1 = 0
becomes
n -
(1/2)n - 1 -
(1/4)n - (3/2) -
(1/8)n - (7/4) -
(1/16)n - (15/8) -
(1/32)n - (31/16) -
(1/64)n - (63/32) -
(1/128)n - (96/64) = -1
Multiply everything by 128 to get rid of all the fractions, so this becomes

128n - 64n - 128 - 32n - 192 - 16n - 224 - 8n - 240 - 4n - 248 - 2n - 252 - n - 254 = -128

n - 1538 = -128
n = 1666

Proof : All I did was plug in 1666 for the value of n, and here is what i got. By the way, if you are wondering where did i get 834, because half of 1666 is 833 instead of 834, don't forget to add 1 more each time.

1666 - 834 - 418 - 210 - 106 - 54 - 28 - 15 + 1 = 0

If you work out the math from 1666 to 15, you will get 1 apple left, and according to the problem, when you finished, you had one apple left.