To solve for the least positive and greatest negative integer values of
k in the expression
(1+i1−i)k=−i, follow these steps:
Start with the given equation:
(1+i1−i)k=−iSimplify the fraction
1+i1−i :
1+i1−i⋅1−i1−i=(1+i)(1−i)(1−i)2Calculate the numerator and the denominator:
(1−i)2=1−2i+i2=1−2i−1=−2i(1+i)(1−i)=1+i−i−i2=1+1=2Substitute back into the fraction:
(1+i)(1−i)(1−i)2=2−2i=−iTherefore, the expression simplifies to:
(−i)k=−iFor
(−i)k=−i, it implies:
k≡1(mod4)From this, determine the least positive integer
m and greatest negative integer
n :
m=1(sincek=1is the smallest positive integer solution)n=−3 (since
k=−3 satisfies the condition
−i in the sequence and is the largest negative solution)
Calculate
m−n :
m−n=1−(−3)=4Thus,
m−n=4.