Sums of Powres of the Imaginary Unit i

Examples on sums of powers of imaginary unit \( i \).

Example 1


The imaginary number $i$ is defined such that $i^2=−1$. What does $i−i^2+i^3−i^4+i^5−i^6...−i^9$ equal?

  1. $-i$
  2. $-1$
  3. $1$
  4. $i$
  5. $0$

Solution


  1. We first rewrite the above sum as follows
    $i−i^2+i^3−i^4+i^5−i^6...i^9=(i+i^3+i^5+i^7+i^9)-(i^2+i^4+i^6+i^8)$

  2. We use the the properties of $i$ given by $i^3=-i$, $i^5=i$, $i^7=-i$ and $i^9=i$ to simplify the sum $i+i^3+i^5+i^7+i^9$

    $i+i^3+i^5+i^7+i^9 = i-i+i-i+i=i$

  3. We use the the properties of $i$ given by $i^2=-1$, $i^4=1$, $i^6=-1$ and $i^8=1$ to simplify the sum $i^2+i^4+i^6+i^8$

    $i^2+i^4+i^6+i^8 = -1+1-1+1=0$

  4. Hence

    $i−i^2+i^3−i^4+i^5−i^6...i^9= i+0=i$

    Answer D