Question

calculati
$i*i^2*i^3*...*i^{2013}$

Answer 1

$i\cdot i^2\cdot i^3\cdot ...\cdot i^{2013} = \\ \\=i^1\cdot i^2\cdot i^3\cdot ...\cdot i^{2013}\\ \\ = i^{1+2+3+...+2013} = \\ \\ = i^{\dfrac{2013\cdot (2013+1)}{2}} = \\ \\ =i^\dfrac{2013\cdot 2014}{2}} = \\ \\ =i^\big{2013\cdot 1007}$

Avem un ciclu:

$i^1 = i\\ i^2 = -1 \\ i^3 = -i \\ i^4 = 1 \\ ------ \\ i^5 = i \\ i^6 = -1 \\ i^6 = -i \\ ......$

Avem niste conditii, pentru i^n;

1) Daca n impar si:
                a) Daca (n+1)/2 impar  => i^n = i
                b) Daca (n+1)/2 par  => i^n = -i

2) Daca n par si:
               a) Daca n/2 impar  => i^n = -1
               b) Daca n/2 par  => i^n = 1

Noi avem:
i^{2013×1007}

U2 (2013×1007) = U2 (13×7) = 91           (U2 -> ultimele doua cifre)

   91 impar:
           (91+1)/2 = 46 par  => i^(2013×1007) = -i

$\Rightarrow \boxed{i\cdot i^2\cdot i^3\cdot ...\cdot i^{2013} = -i}$