Question 194943
Since {{{i=sqrt(-1)}}}, this means that 



{{{i^1=i}}}



{{{i^2=(sqrt(-1))^2=-1}}} or {{{i^2=-1}}}



{{{i^3=(sqrt(-1))^3=(sqrt(-1))^2*sqrt(-1)=-1*i=-i}}} or {{{i^3=-i}}}



{{{i^4=(sqrt(-1))^4=(sqrt(-1))^2*(sqrt(-1))^2=-1*-1=1}}} or {{{i^4=1}}}



{{{i^5=(sqrt(-1))^5=(sqrt(-1))^4*sqrt(-1)=1*i=i}}} or {{{i^5=i}}}



etc...


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So for every increment of the exponent, the powers of "i" are: i, -1, -i, 1, i, etc... and this pattern repeats every 4 terms (notice how the 5th term is identical to the first term). 



This pattern can be generalized to the following:


For {{{i^k}}} (where "k" is a whole number), 



If the remainder of {{{k/4}}} is 0, then {{{i^k=1}}}.



If the remainder of {{{k/4}}} is 1, then {{{i^k=i}}}.



If the remainder of {{{k/4}}} is 2, then {{{i^k=-1}}}.



If the remainder of {{{k/4}}} is 3, then {{{i^k=-i}}}.



So out at the 100th term, since {{{100/4=25}}} remainder 0, this means that {{{i^(100)=1}}}