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Question
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20.
Solution
1 + i2 + i4 + i6 + i8 + ... + i20
= 1 + (i2 + i4) + (i6 + i8) + (i10 + i12) + (i14 + i16) + (i18 + i20)
= 1 + [i2 + (i2)2] + [(i2)3 + (i2)4] + [(i2)5 + (i2)6] + [(i2)7 + (i2)8] + [(i2)9 + (i2)10]
= 1 + [– 1 + ( – 1)2] + [(– 1)3 + (– 1)4] + [(– 1)5 + (– 1)6] + [(– 1)7 + (– 1)8] + [(– 1)9 + (– 1)10] ...[∵ i2 = – 1]
= 1 + (– 1 + 1) + (– 1 + 1) + (– 1 + 1) + (– 1 + 1) + (– 1 + 1)
= 1 + 0 + 0 + 0 + 0 + 0
= 1.
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