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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