Algebra Word Problem!!! Problem of the Week, I can use any sources! Plz help!?

Question

Albert spent all his money in 5 stores. In each store, he spent $1 more than half of what he had when he entered that store. How much money did Albert have when he entered the first store?

I have weekly Problem of the Weeks in algebra 1 class. My teacher lets us use ANY source at all. So technically, I am not cheating or anything by putting the question an yahoo! answers. I have tried to get the answer but I am really puzzled.

Answer 1

Can we assume that he had no money at the end of all his shopping?

To solve this, I would work backward. Let's assume he ends with $0.

In general, if he starts with S dollars in a store, he spends S/2 + 1.
In an equation this would be:
S - (S/2 + 1) = E

State this in terms of S:
S - S/2 - 1 = E
S/2 - 1 = E
S/2 = E + 1
S = 2(E+1)

So in other words, start with his ending amount, add 1 and double.

If he ended at the 5th store with $0, then he started in the store with 2(0+1) = $2.

If he ended at the 4th store with $2, then he started in the store with 2(2+1) = $6

If he ended at the 3rd store with $6, then he started in the store with 2(6+1) = $14

If he ended at the 2nd store with $14, then he started in the store with 2(14+1) = $30

If he ended at the 1st store with $30, then he started in the store with 2(30+1) = $62

So assuming he ends with $0, he must have started with $62

To confirm this, walk through the stores again in order.
He has $62, spends $32, leaving $30.
Then he spends $16, leaving $14
Then he spends $8, leaving $6
Then he spends $4, leaving $2
Finally he spends $2, leaving nothing.

(*Note if you had a different ending amount, just use the same logic of adding 1 and doubling.)

Answer 2

We know that he uses 1/2x + 1 in each store, be x the amount he has when he entered the store.

Amount he has before he visited the 5th store:
x - 1/2 x - 1 = 0
1/2 x = 1
x = 2

Amount he has before he visited the 4th store:
x - 1/2 x - 1 = 2
1/2 x = 3
x = 6

Amount he has before he visited the 3rd store:
x - 1/2 x - 1 = 6
1/2 x = 7
x = 14

Amount he has before he visited the 2nd store:
x - 1/2 x - 1 = 14
1/2 x = 15
x = 30

Amount he has before he visited the 1st store:
x - 1/2 x - 1 = 30
1/2 x = 31
x = 62

Albert has $62 when he entered the first store.

Answer 3

You can easily find a pattern.
Let x be the amount of money to be found.
(1/2)((((1/2)(((1/2)((1/2)(x - x/2 - 1) - 1) - 1) - 1) -1 = 0
Solve for x,
x = (2(2(2(2(2*1+1)+1)+1)+1)
x = $62
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Ideas: You go from inside to outside to get, 2, 6, 14, 30, 62.