prove:

Use M.I.:

 

Let n = 1:

 

i^(2n + 1) = i^3 = -i........

 

The question is not correct...... Do you mean i^(4n + 1) = i?

i^(2 n + 1) = i  for all positive integers n

 

\(i^1=i\\ i^2=-1\\ i^3=-i\\ i^4=+1\\ i^5=i\\ ...\\ i^{1+4k}=i\)

 

 

 

2n+1=4k+1     where n and k are both positive integers

2n    = 4k

n      = 2k

n = 2,4,6, .....   all positive even integers