Hint:
You're trying to find \(z^{53}+z^{52}+z^{51}+z^{50}+z^{49}\), so factor out \(z^{49}\) to get \(z^{49}(z^4+z^3+z^2+z+1)\). We know what \(z^2+z+1\) is, so substitute it in. You now have\(z^{49}(z^4+z^3)\), which can be written as \(z^{51}(z^2+z)\). Now, we know that \(z^2 + z + 1=0\), so \(z^2+z=-1\). Substituting, we get \(-z^{51}\).
Continue from here.
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+62 