I need a certain type of formula?

Question

I have an item which has an increasing price of 10% each time you buy one.
I need to find a formula to find the total cost of a certain number of items (any number)
The initial cost is 2000, so it increases by 200 each time.
The cost of 6 is 15000.
What is a formula that connects the two variables? (total cost and amount purchased)

jbdeeds · Accepted Answer

I believe you are talking about an arithmetic price increase -- where you add on 200 for each purchase.  This is not really an ongoing 10% price increase, which would multiply by 1.1 for each purchase.  The example of the cost of 6 confirms this.  So your progression of total costs is:

Buy 1:  2000
Buy 2:  2000 + (2000 + 200)
Buy 3:  2000 + (2000 + 200) + (2000 + 2*200)
...
Buy n: 2000 + (2000 + 200) + (2000 + 2*200) + ...+ (2000 + (n-1)*200)

In the general sum, there are n occurences of 2000.  If we collect these together and look at what remains, we see 200 is a common factor.  So the sum can be written:
Buy n:  2000*n + 200(1 + 2 + 3 + ...(n-1))

To get a final closed form, we invoke the well-known formula for the sum of the integers from 1 to n-1.  That value is n*(n-1)/2.  So your final formula for the purchase of n items is:

2000* n + [200*(n*(n-1)/2].

For n = 1, this gives 2000.
For n = 6, this gives 12000 + 3000 = 15000
For n = 10, this gives 20000 + 200*10*9/2 = 29000, etc.

I need a certain type of formula?

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