Question

solve above question

Answer 1

★ COMPLEX NUMBERS ★

$i = \sqrt{ - 1 } \\ {i}^{2} = - 1 \\ {i}^{3} = - i \\ i {}^{4} = \: \: 1 \\ \\ {i}^{2n} + i {}^{2n + 1} + i {}^{2n + 2} + {i}^{2n + 3} \\ = \: ( - 1) {}^{n} + (( - 1) {}^{n} .i) + (( - 1) {}^{n} .( - 1)) + (( - 1) {}^{n} .( - i)) \\ \\ = ( - 1) {}^{n} (1 + i - 1 - i) \\ \\ = ( - 1) {}^{n} \times 0 \\ \\ = 0 \: ans.$