Question

Simplify

${i}^ {n + 100} + {i}^{n + 50} + {i}^{n + 48} + { i}^{n + 46}$
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Answer 1

${i}^{n + 100} + {i}^{n + 50} + {i}^{n + 48} + {i}^{n + 46} \\ = {i}^{n} . {i}^{100} + {i}^{n} . {i}^{50} + {i}^{n} . {i}^{48} + {i}^{n} . {i}^{46} \\ = {i}^{n} .( {i}^{4} )^{25} + {i}^{n} .( {i}^{4} )^{12} .{i}^{2} + {i}^{n} .( {i}^{4} ) ^{12} + {i}^{n} . ({i}^{4} ) ^{11} . {i}^{2} \\ = {i}^{n} + {i}^{n} .( - 1) + {i}^{n} + {i}^{n} .( - 1) \\ = {i}^{n} - {i}^{n} + {i}^{n} - {i}^{n} \\ = 0$

[since i⁴=1]