If z 2 + z + 1 = 0 , z ∈ C , then ∑ n = 1 15 z n + ( - 1 ) a 1 z n 2 is equal to _____.

If z2+z+1=0,z∈C, then ∑n=115zn+(-1)a1zn2 is equal to _____.

Solution

Given, z2+z+1=0⇒z=w,w2

Now, ∑n=115zn+-11zn2=∑n=115z2n+1z2n+2-1n

=∑n=115w2n+1w2n+2-1n

=w21-w301-w2+1w21-1w301-1w2+2-1

=w21-11-w2+1w21-11-1w2-2

=0+0-2=2

Asked in: JEE Main 2022 (26 Jun Shift 2)

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