For distinct complex numbers z1,z2,…,zn, the value of z2−z12+z3−z22+…..+zn−zn−12zn−z12 can not be less than

zn−z1=zn−zn−1+…+z3−z2+z2−z1⇒zn−z1≤zn−zn−1+zn−1−zn−2+…+z3−z2+z2−z1⇒z2−z12+z3−z22+…+zn−zn−12(n−1)2≥zn−z12(n−1)2≥zn−zn−1+zn−1−zn−2+…+z2−z1(n−1)2⇒z2−z12+z3−z22+…+zn−zn−12zn−z12≥1n−1.

For distinct complex numbers z1,z2,…,zn, the value of z2−z12+z3−z22+…..+zn−zn−12zn−z12 can not be less than

zn−z1=zn−zn−1+…+z3−z2+z2−z1⇒zn−z1≤zn−zn−1+zn−1−zn−2+…+z3−z2+z2−z1⇒z2−z12+z3−z22+…+zn−zn−12(n−1)2≥zn−z12(n−1)2≥zn−zn−1+zn−1−zn−2+…+z2−z1(n−1)2⇒z2−z12+z3−z22+…+zn−zn−12zn−z12≥1n−1.

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For distinct complex numbers $z_{1}, z_{2}, \dots, z_{n}, the value of \frac{{|z_{2} - z_{1}|}^{2} + {|z_{3} - z_{2}|}^{2} + \dots . . + {|z_{n} - z_{n - 1}|}^{2}}{{|z_{n} - z_{1}|}^{2}}$ can not be less than

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$\frac{1}{n - 2}$

$\frac{1}{n + 1}$

$\frac{1}{n - 1}$

$\frac{1}{n}$

answer is A.

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Detailed Solution

$\begin{array}{l} z_{n} - z_{1} = z_{n} - z_{n - 1} + \dots + z_{3} - z_{2} + z_{2} - z_{1} \\ \Rightarrow |z_{n} - z_{1}| \leq |z_{n} - z_{n - 1}| + |z_{n - 1} - z_{n - 2}| + \dots + |z_{3} - z_{2}| + |z_{2} - z_{1}| \\ \Rightarrow {\frac{{|z_{2} - z_{1}|}^{2} + {|z_{3} - z_{2}|}^{2} + \dots + {|z_{n} - z_{n - 1}|}^{2}}{(n - 1)})}^{2} \geq \frac{{|z_{n} - z_{1}|}^{2}}{(n - 1)^{2}} \\ \geq ({\frac{|z_{n} - z_{n - 1}| + |z_{n - 1} - z_{n - 2}| + \dots + |z_{2} - z_{1}|}{(n - 1)}|}^{2} \\ \Rightarrow \frac{{|z_{2} - z_{1}|}^{2} + {|z_{3} - z_{2}|}^{2} + \dots + {|z_{n} - z_{n - 1}|}^{2}}{{|z_{n} - z_{1}|}^{2}} \geq \frac{1}{n - 1} . \end{array}$