## Guessing Game

### Problem 406

We are trying to find a hidden number selected from the set of integers {1, 2, ..., `n`} by asking questions.
Each number (question) we ask, we get one of three possible answers:

- "Your guess is lower than the hidden number" (and you incur a cost of
`a`), or - "Your guess is higher than the hidden number" (and you incur a cost of
`b`), or - "Yes, that's it!" (and the game ends).

Given the value of `n`, `a`, and `b`, an *optimal strategy* minimizes the total cost __for the worst possible case__.

For example, if `n` = 5, `a` = 2, and `b` = 3, then we may begin by asking "**2**" as our first question.

If we are told that 2 is higher than the hidden number (for a cost of `b`=3), then we are sure that "**1**" is the hidden number (for a total cost of **3**).

If we are told that 2 is lower than the hidden number (for a cost of `a`=2), then our next question will be "**4**".

If we are told that 4 is higher than the hidden number (for a cost of `b`=3), then we are sure that "**3**" is the hidden number (for a total cost of 2+3=**5**).

If we are told that 4 is lower than the hidden number (for a cost of `a`=2), then we are sure that "**5**" is the hidden number (for a total cost of 2+2=**4**).

Thus, the worst-case cost achieved by this strategy is **5**. It can also be shown that this is the lowest worst-case cost that can be achieved.
So, in fact, we have just described an optimal strategy for the given values of `n`, `a`, and `b`.

Let $C(n, a, b)$ be the worst-case cost achieved by an optimal strategy for the given values of `n`, `a` and `b`.

Here are a few examples:

$C(5, 2, 3) = 5$

$C(500, \sqrt 2, \sqrt 3) = 13.22073197\dots$

$C(20000, 5, 7) = 82$

$C(2000000, \sqrt 5, \sqrt 7) = 49.63755955\dots$

Let $F_k$ be the Fibonacci numbers: $F_k=F_{k-1}+F_{k-2}$ with base cases $F_1=F_2= 1$.

Find $\displaystyle \sum \limits_{k = 1}^{30} {C \left (10^{12}, \sqrt{k}, \sqrt{F_k} \right )}$, and give your answer rounded to 8 decimal places behind the decimal point.