There was a two part question assigned to my class for homework. It's already been turned in, but I couldn't get the 2nd part, and am curious how it is done:

part a:

suppose there is a k between 0 and n such that either:

(i) $\displaystyle C_0, C_1, ..., C_k <= 0$ and $\displaystyle C_(k+1), C_(k+2),..., C_(n)>= 0$

or

(ii) $\displaystyle C_0, C_1, ..., C_k >=0$ and $\displaystyle C_(k+1), C_(k+2),..., C_(n) <= 0$

show there is a unique i > -1 for which the net present value of this transaction is 0.

this is the part I got. I don't need help on it, but am just introducing it, because I'm sure it somehow is used to prove the next part.

part b:

Let $\displaystyle C_0, C_1, ..., C_n$ be an arbitrary sequence of net cashflows, and let

$\displaystyle

F_0 = C_0,

F_1 = C_0 + C_1,

.

.

.

F_n = C_0 + C_1 + ... + C_n

$

Suppose both $\displaystyle F_0$ and $\displaystyle F_n$ are non-zero, and that the sequence $\displaystyle {F_0, F_1, ..., F_n}$ has exactly one change of sign.

Show there is a unique i > 0 such that the net present value of these cash flows is 0 (although there may be one or more negative roots).

This is the part I need help on. I can show what I've done so far, but it really isn't anything except showing thereisa root i > 0. I haven't shown that it is unique.

my work:

we need to show that the equation has a unique solution 0 < v < 1.

if v = 0,

if v = 1,

we are given F_0 and F_n are non-zero and opposite signs of each other, so by the intermediate value theorem there exists a root v, 0 < v < 1.

This is all I have come up with, I don't know if that is a good starting point for the rest of the solution or not. any help would be appreciated! Thanks in advance.