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?

• what do you mean with "sum over $f(x)$"? – Arnaldo Mar 22 '17 at 14:02
• I mean for example at x=10 the sum over f(x) should be f0) + f(1) + ... + f(10) @Arnaldo – m.awad Mar 22 '17 at 14:41

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.

Let $$g(x) = f(0) + f(1) + f(2) + ... + f(x) \\ = \sum_{n=0}^{x} f(n)$$

Since $f(x) = 2x$, $$\\ g(x) = \sum_{n=0}^{x} 2n = 2\sum_{n=0}^{x} n \\ = 2\sum_{n=1}^{x} n = 2(1+2+...+x)$$

There is a useful known result: $$1+2+...+N= \sum_{n=1}^{N} n = \frac{N(N+1)}{2}$$

So, $$g(x) = 2\sum_{n=1}^{x} n = 2 \Big(\frac{x(x+1)}{2} \Big) = x(x+1)$$

Thus your cumulative sum function may be written as

$$g(x) = x^2+x$$

• This answers the OP's algebra question correctly, but it does not answer his real question, since he didn't turn that question into algebra correctly. – Ethan Bolker Mar 22 '17 at 18:06