find equation of a curve that represents the sum of f(x) Given a straight line with equation f(x)=2x where x belongs to {0,1,2,3,4,5}, 
how do i find g(x) (a curve?) which represents the sum over f(x)
In plain english if the price of an object doubles every time I purchase it, what will be the total cost if I purchased it 10 times?

Your plain English question makes much more sense than your attempt to turn it into algebra.
You should be able to guess the pattern here:
number purchased        total price
       1                           P
       2                P + 2P =  3P
       3               3P + 4P =  7P
       4               7P + 8P = 15P
      ...

Hint. Think about nearby powers of $2$ in the last column.
