Question

Find the value of : (-1)^n +(-1)^2n+(-1)^2n+1 +(-1)^4n+1 where n is any
positive odd integer.​

Answer 1

$(-1)^n+(-1)^{2n}+(-1)^{2n+1}+(-1)^{4n+1}\\= -1+1-1-1\\=-2$

(1) We know that if the power of a negative number odd then the number comes out to be negative. For example,

$(-1)^1=-1\\(-1)^3=-1$

(2) If the power of a negative number is 2n where n is odd then as the power gets multiplied by 2 it gives a even number and (-1) becomes 1.

$(-1)^{2*1}=(-1)^2=1\\(-1)^{2*3}=(-1)^6=1$

(3) As, 2n gives a positive number but when it is added with 1 it becomes odd.

$(-1)^{2*1+1}=(-1)^{2+1}=(-1)^3=-1\\(-1)^{4n+1}=(-1)^{4*1+1}=(-1)^5=-1$