# This could not be more imaginary

Algebra Level 5

Let $$f(n) = { a }_{ 1 }^{ n }+{ a }_{ 2 }^{ n }+{ a }_{ 3 }^{ n }$$.

Given that

${ f(1)=a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }=0\\ f(2)={ a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 }=i\\ f(3)={ a }_{ 1 }^{ 3 }+{ a }_{ 2 }^{ 3 }+{ a }_{ 3 }^{ 3 }={ i }^{ i }$

and that $-(f(4)+f(7))=\frac { a+b\left( { e }^{ \frac { \pi }{ c } } \right) }{ d }$

Find $$a+b+c+d$$ given that they are integers $.$$.$ $$\textbf{Assumptions}$$

$$\bullet$$ $$i=\sqrt { -1 }$$

$$\bullet$$ $$(e)$$ is Euler's number or $$\left(e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }\right)$$

$.$$.$

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