# Math Help - Arithmetic Series

1. ## Arithmetic Series

How would one put the following question in an equation?
A clothing store ordered 300 swearters and sold 20 of them at $100 each in the first week. In the second week, the selling price was lowered by$10, and 40 sweaters were sold. In the third week, the selling price was lowered by another \$10, and 60 sweaters were sold. If the pattern continued,

a) how many week did it take to sell all the sweaters?
b) what was the selling price in the final week?

For a), I initially tried doing this:

0=300-20*(n-1)
=300-20n+20
n=15

but that didn't turn out right, so I'm starting to speculate a recusive aspect to the amount of sweaters sold? Same goes for part b). Any help here?

EDIT: I later tried this for a), it seems closer, but still wrong:

300=(n/2)*(20*(n-1))
600=n*(20n-20)
0=20n^2-20n-600
=20(n^2-n-30)
n=-5,6

The answer to a) is supposed to be 5, but this is as close as I can get.

2. I dislike these questions because it's so easy to make a mistake.

The formula for an arithmetic series is

$\sum n =\frac{n}{2}[2a+(n-1)d]$

Where:
n = number of terms in the sequence that you are summing
a = first term in the sequence
d = the rate that the sequence is increasing or decreasing

In a), we can ignore prices. It's just the amount of sweaters that matters.

On the first week, 20 sweaters were sold. So a=20.
The amount of sweaters sold then increases by 20 to 40. So d=20.
And we want the sum of sweaters sold to be equal to 300
In this case, n=number of weeks

So the formula becomes:
$300=\frac{n}{2}[2(20)+(n-1)(20)]$

By simplifying and manipulating, I got the correct answer of n=5 doing this, do you get the same result?

If yes, then work out what needs to be changed for b).

I can help further if necessary.

3. Thanks for the clarification. It made it a lot more sense. Forgive my careless mistakes.

4. Upon review, what you did second was nearly correct, you seem to have made a signs error somewhere along the line. But hey, we all do that sometimes...