A dress is listed at $150. At the end of each week the price is reduced by 10%.

(a) What is the price of the dress after 8 weeks?(b) After how many weeks will the dress be less than $50?

(hint: If 10% of the price is taken off, then 90% of the price remains.)

Answer

After one week, the price will be reduced by 10%. This means the dress will cost 90% of its original price, so it will cost

0.9(150) = 135.

After two weeks the price will be reduced by 10% again, so the dress will cost

0.9(135) = (0.9)(0.9)(150).

Continuing in this fashion, after *n* weeks the price of the dress will be

*a*_{n} = (.9)^{n}(150).

(a) After 8 weeks, the price of the dress will be

*a*_{8} = (0.9)^{8}(150) = 64.57.

The dress will cost $64.57.

(b) This question is asking for what value of *n* we have *a*_{n}<50.

At this point it's tempting to take logs of both sides.

If we do that, we get

*n* < 10.4.

That doesn't make sense, because we're looking for a statement of the form

*n* > ...

The weirdness is because is negative, and negative signs do funny things to inequalities. To avoid the negatives, let's rearrange the fractions before taking the log.

Since *n* must be a whole number, we can conclude that after *n* = 11 weeks the dress will be less than $50.