If is a complex cube root of unity, then ^ ( 1 3 + 2 9 + 4 27 + ) + ^ ( 1 2 + 3 8 + 9 32 + ) is equal to

Step-by-step solution

We have, ^ ( 1 3 + 2 9 + 4 27 + . ) + ^ ( 1 2 + 3 8 + 9 32 + . ) = ? Here, 1 3 + 2 9 + 4 27 + Follows infinite G.P. series so its sum, aligned & S_ = a 1-r = 1 3 1- 2 3 & = 1 3 3 1 =1 ^ ( 1 3 + 2 9 + 4 27 + ) = aligned Now, 1 2 + 3 8 + 9 32 + . .+ also follows infinite G.P. series so its sum aligned & S _ = a 1-r = 1 2 1- 3 4 = 1 2 4 1 =2 & ^ ( 1 2 + 3 8 + + ) = ^2 & Hence, ^ ( 1 3 + 2 9 + + ) + ^ ( 1 2 + 3 8 + + ) = + ^2=-1 & [ ^2+ +1=0 ^2+ =-1 ] & aligned