If $f(x)$ is a 5th degree monic polynomial such that :

$f(1) = 1$

$f(2) = 2$

$f(3) = 5$

$f(4) = 13$

$f(5) = 34$

then find the value of $(f(6) + f(7) + f(8))$

**
This problem is original
**

The answer is 3943.

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Method of differences:

$\begin{array}{ccccccccccccccc} 1 && 2 && 5 && 13 && 34 && 201 && 892 && 2850 \\ & 1 && 3 && 8 && 21 && 167 && 691 && 1958 & \\ && 2 && 5 && 13 && 146 && 524 && 1267 && \\ &&& 3 && 8 && 133 && 378 && 743 &&& \\ &&&& 5 && 125 && 245 && 365 &&&& \\ &&&&& 120 && 120 && 120 &&&&& \end{array}$

Because this is a fifth degree polynomial, all 5th differences are equal. Because it is monic, the 5th differences are equal to $5! = 120$ .

Solution: $201 + 892 + 2850 = \boxed{3943}$ .