Question

1+i^10+i^100-i^1000=0​

Answer 1

Answer: The proof is done below.

Step-by-step explanation:  We are given to prove the following equality involving the imaginary number i :

$1+i^{10}+i^{100}-i^{1000}=0$

We know that the value of the imaginary number i is as given as

$i=\sqrt{-1}.$

So, we get

$i^{10}=(i^2)^5=(-1)^5=--1,\\\\\\i^{100}=(i^2)^{50}=(-1)^{50}=1,\\\\\\i^{1000}=(i^2)^{500}=(-1)^{500}=1.$

Therefore,

$L.H.S.\\\\=1+i^{10}+i^{100}-i^{1000}\\\\=1+(-1)+1-1\\\\=1-1+1-1\\\\=0\\\\=R.H.S.$

Hence proved.