The maximum value of n < 101 , such that 1+ _ k-1 ^ n i^ k =0 is

Step-by-step solution

We have, _ n=1 ^ n i^ k =i+i²+i³+i⁴+i⁵+i⁶+i⁷+i⁸+i⁹+ Now as, i⁴=i⁸=i¹²= i^ n =1 and i+i²+i³+i⁴=0 So, _ n=1 ^ n i^ k =i+i²+i³+i⁴+i+i⁶+ i⁷+i⁸+i²+ aligned &=i+i²+i³+i⁴+ (i+i²+i³+i⁴ )+ (i+i²+i³+i⁴ )+ &=0+0+0+ aligned When we take n=99 after 96 th term, we will be left with 0+i+i²+i³=-1 So , for 1+ _ k=1 ^ n i^ k =0