Question

Care este partea reala a produsului i^2*i^4*i^6*...*i^2014?

Answer 1

[tex]i^2\cdot i^4\cdot i^6\cdot\ldots\cdot i^{2014}=\\ i^{2+4+6+\ldots+2014}=i^{2\cdot \frac{2014\cdot 2015}{2}}=i^{2014\cdot 2015}= i^{4058210}\\ \text{Stim ca:}\\ i^{4k}=1\\ i^{4k+1}=i\\ i^{4k+2}=-1\\ i^{4k+3}=-i\\ 4058210:4=1014552,rest=2\Rightarrow i^{4058210}=-1\\ \text{Partea reala este -1}. [/tex]