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

Question

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

MARKS App · Accepted Answer

Given, z 2 + z + 1 = 0 ⇒ z = w , w 2 Now, ∑ n = 1 15 z n + - 1 1 z n 2 = ∑ n = 1 15 z 2 n + 1 z 2 n + 2 - 1 n = ∑ n = 1 15 w 2 n + 1 w 2 n + 2 - 1 n = w 2 1 - w 30 1 - w 2 + 1 w 2 1 - 1 w 30 1 - 1 w 2 + 2 - 1 = w 2 1 - 1 1 - w 2 + 1 w 2 1 - 1 1 - 1 w 2 - 2 = 0 + 0 - 2 = 2

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