If is the imaginary cube root of unity such that | ( _ r =1 ^ n ( r _ p =1 ^ r ( ^ P -1 ) ) )-155 |= ( _ r =1 ^ n ( r _ p =1 ^ r ^ P -1 ) )-155 , then n is equal to -

Step-by-step solution

_ r=1 ^ n ( r _ p =1 ^ r ( ^ P -1 ) )-155 = _ r =1 ^ n ( r ( ^0+ ^1+ ^2+ ^2+ . .+ ^ r -1 ) )-155 =1 ^0+2 ( ^0+ ^0 )+3 ( ^0+ ^1+ ^2 )+4 ( ^0+ ^1+ ^2+ ^3 ) . n ( ^0+ ^1+ ^2+ + ⁴⁻¹ )-155 =1+2 (- ^2 )+3(0)+4(1)+5 (- ^2 )+ . upto n terms -155 will be real if n =29,30,31