Question

i^107+i¹¹²+i^117+i^122=0
prove : ​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:i^{107}\:+\:i^{112}\:+\:i^{117}\:+\:i^{122}\:=\:0}}}$

Step-by-step-explanation:

The given equation is

$\displaystyle{\sf\:i^{107}\:+\:i^{112}\:+\:i^{117}\:+\:i^{122}\:=\:0}$

We have to prove this equation.

We know that,

$\displaystyle{\sf\:i\:=\:\sqrt{-\:1}}$

$\displaystyle{\sf\:i^2\:=\:-\:1}$

$\displaystyle{\sf\:i^3\:=\:-\:i}$

$\displaystyle{\sf\:i^4\:=\:1}$

$\displaystyle{\sf\:i^{4n}\:=\:1\:\quad\cdots\:n\:\in\:Z}$

Now,

$\displaystyle{\sf\:i^{107}\:+\:i^{112}\:+\:i^{117}\:+\:i^{122}\:=\:0}$

$\displaystyle{\implies\sf\:LHS\:=\:i^{107}\:+\:i^{112}\:+\:i^{117}\:+\:i^{122}}$

$\displaystyle{\implies\sf\:LHS\:=\:i^{104\:+\:3}\:+\:i^{4\:\times\:28}\:+\:i^{116\:+\:1}\:+\:i^{120\:+\:2}}$

We know that,

$\displaystyle{\pink{\sf\:a^{m\:+\:n}\:=\:a^m\:\times\:a^n}}$

$\displaystyle{\implies\sf\:LHS\:=\:i^{104}\:\times\:i^3\:+\:1\:+\:i^{116}\:\times\:i^1\:+\:i^{120}\:\times\:i^2}$

$\displaystyle{\implies\sf\:LHS\:=\:i^{4\:\times\:26}\:\times\:(\:-\:i\:)\:+\:1\:+\:i^{4\:\times\:29}\:\times\:i\:+\:i^{4\:\times\:30}\:\times\:(\:-\:1\:)}$

$\displaystyle{\implies\sf\:LHS\:=\:1\:\times\:(\:-\:i\:)\:+\:1\:+\:1\:\times\:i\:+\:1\:\times\:(\:-\:1\:)}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:i\:+\:1\:+\:i\:-\:1}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:i\:+\:i\:+\:1\:-\:1}$

$\displaystyle{\implies\sf\:LHS\:=\:0\:+\:0}$

$\displaystyle{\implies\sf\:LHS\:=\:0}$

$\displaystyle{\sf\:RHS\:=\:0}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:LHS\:=\:RHS\:}}}}$

Hence proved!

Answer 2

Answer:

topic :

• E and I

given :

• i^107+i¹¹²+i^117+i^122=0

solution :

• please check the attached file there wil you find your answer