SOLUTION: A dry cleaner charges 5 dollars to clean one item. She offfers to clean to clean a second item for 4.50 and a third item for 4.00. Which item will be cleaned or free? I say its 11

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Question 802552: A dry cleaner charges 5 dollars to clean one item. She offfers to clean to clean a second item for 4.50 and a third item for 4.00. Which item will be cleaned or free?
I say its 11 because its
4.00 for 3 items
3.50 for 4 items
3.00 for 5 items
2.50 for 6 items
2.00 for 7 items
1.50 for 8 items
1.00 for 9 items
.50 for 10 items
11th item is no charge
BUT 5 dollars ÷ 50 cents = 10 items
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You are correct. If that pattern continues, the 11th item would be free.

THE FIFTH GRADER WOULD SAY:
Each additional item costs $0.50 less to clean.
To get one item cleaned for free, you need to include 10 additional items along with your first item, for a total of highlight%2811%29 items.

THE HIGH SCHOOL MATH TEACHER WOULD SAY:
It's an arithmetic sequence.
a%5B1%5D=5= price for cleaning the first item, in $
a%5Bn%5D= price for cleaning the nth item, in $
d=-0.5= the common difference, in $
a%5Bn%2B1%5D=a%5Bn%5D%2Bd the recursive formula is a%5Bn%2B1%5D=a%5Bn%5D-0.5
a%5Bn%5D=a%5B1%5D%2B%28n-1%29d in this case is
a%5Bn%5D=5%2B%28n-1%29%28-0.5%29 --> a%5Bn%5D=5-0.5%28n-1%29
If the nth item is free, a%5Bn%5D=0, so
0=5-0.5%28n-1%29 --> 0.5%28n-1%29=5 --> n-1=5%2F0.5 --> n-1=10 --> n=10%2B1 --> highlight%28n=11%29