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Question
Simplify the following:
`sum_("n" = 1)^10 "i"^("n" + 50)`
Sum
Solution
`sum_("n" = 1)^10 "i"^("n" + 50) = sum_("n" = 1)^10 "i"^"n" * "i"^50`
= `sum_("n" = 1)^10 "i"^"n" * "i"^48 * "i"^2`
= `- 1[sum_("n" = 1)^10 "i"^"n"]`
= `- 1[("i" + "i"^2 + "i"^3 + "i"^4) + ("i"^5 + "i"^6 + "i"^7 + "i"^8) + "i"^9 + "i"^10]`
= `- 1[0 + 0 "i" + "i"^2]`
= `- 1("i" - 1)`
= 1 – i
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Introduction to Complex Numbers
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