Question

Find the value of 1 + io + i20 + 130​

Answer 1

We have to find out the value of :- 1 + i¹⁰ + i²⁰ + i³⁰

Where i = √(-1)

$\longrightarrow \rm 1 + {i}^{10} + {i}^{20} + {i}^{30} \\ \\ \\ \longrightarrow \rm 1 + {( {i}^{2} )}^{5} + {( {i}^{2} )}^{10} + {( {i}^{2} )}^{15}$

We know that :- i² = -1

$\longrightarrow \rm 1 + {( - 1 )}^{5} + {( - 1 )}^{10} + {( { - 1} )}^{15} \\ \\ \\ \longrightarrow \rm 1 + {( - 1 )} + {( 1 )} + {( { - 1} )}\\ \\ \\ \longrightarrow \rm 1 - 1 + 1- 1\\ \\ \\ \longrightarrow \rm 2 - 2\\ \\ \\ \longrightarrow \rm 0$

Answer : 0

Additional Information :-

Properties of Modules :-

If X = a+bi then ,

1) | X | = |- X |

2) | X₁ X₂ | = | X₁ | | X₂ |

3) | X₁ / X₂ | = | X₁ | / | X₂ |

4) | X₁ + X₂ | ≠ | X₁ | + | X₂ |

Properties of argument :-

1) Arg(0) = not defined

2) If X is purely imaginary number then , arg(X) = ± (π/2)

3) Arg( X₁ X₂) = Arg( X₁ ) + Arg( X₂) + 2mπ

4) Arg( X₁ - X₂) = Arg( X₁ ) - Arg( X₂) + 2mπ

5) Arg( Xⁿ) = n Arg( Xⁿ) + 2mπ