SOLUTION: Prove that if k is an integer then (4k+1)i^4k + (4k+2)i^4k+1 + (4k+3)i^4k+2 + (4k+4)i^4k+3= 2-2i. Use this to prove that 1+2i+3i^2+4i^3+...+1995i^1994+1996i^1995=-998-998i

Question 1050433: Prove that if k is an integer then (4k+1)i^4k + (4k+2)i^4k+1 + (4k+3)i^4k+2 + (4k+4)i^4k+3= 2-2i.
Use this to prove that
1+2i+3i^2+4i^3+...+1995i^1994+1996i^1995=-998-998i
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!

===>,
, and
.
===>
=
= .

Now substitute the values of k = 0, 1, 2, 3,...,497, 498 into the formula.
On the left side of the equation, we would get
+...+,
while on the right side, we would get
499*(-2 - 2i) = -998 - 998i.
Therefore,
+...+.

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