If w is an imaginary cube root of unity, then the value of (1+ ) (1+ ^2 ) (1+ ^3 ) (1+ ^4 ) (1+ ^ 3 n )

Step-by-step solution

We have, aligned & (1+ ) (1+ ^2 ) (1+ ^3 ) (1+ ^4 ) (1+ ^ 3 n ) & = [ (1+ ) (1+ ^2 ) (1+ ^4 ) (1+ ^5 ) . & (1+ ^ 3 n-2 ) (1+ ^ 3 n-1 ) (1+ ^3 ) (1+ ^6 ) & . (1+ ^9 ) (1+ ^ 3 n ) ] aligned aligned & =2^n [ (- ^2 )(- ) (- ^2 )(- ) ] (- ^2 )(- ) ] & [(1+w) (1+w^2 ) (1+w) (1+w^2 ) (1+w) (1+w^2 ) . & . (1+w) (1+w^2 ) ] (1+1)(1+1) (1+1) & =2^n aligned