Question

The principal argument of z = i - 2 - 3i+ 4 + 5i - 6 - 7i+8+...
100 terms, is​

Answer 1

(1) 50(1-i)

(2) 25(1+i)

(3) 100(1-i)

(4) 25i

Solution:

We know i2 = -1 and i4 = 1

Let S = i – 2 – 3i + 4 +…100i100

= i+ 2i2 + 3i3 + 4i4 + 5i5+….100i100

iS = i2+ 2i3+ 3i4+….+99i100 + 100i101

S-iS = (i + i2 + i3 + i4+…i100)-100i101

S(1-i) = i(i100-1)/(i-1) – 100i101

= -100i101 (since i100 = 1)

S = -100i101/(1-i)

= -100i/(1-i)

= -100i(1+i)/2 (rationalizing)

= -50i(1+i)

= -50(i-1)

= 50(1-i)

Hence option (1) is the answer.