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Question
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
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Solution
`(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")`
= `((1 - 2"i")(3 + 4"i") + (3 - 4"i")(1 + 2"i"))/((3 + 4"i")(3 - 4"i"))`
= `(3 + 4"i" - 6"i" - 8"i"^2 + 3 + 6"i" - 4"i" - 8"i"^2)/(9 - 16"i"^2)`
= `(6 - 16"i"^2)/(9 - 16(-1))`
= `(6 - 16(-1))/(9 + 16)` ...[∵ i2 = – 1]
= `22/25`, which is a real number.
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i2 + i3 + ... + i4000 =
