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MATHMAA

# Proof

Proof of inequation (1/a1+1/a2+.....+1/an)(a1+a2+...+an)>=n^2 if and only if a1=a2=...=an .

Q) Prove that  $$( \frac{1}{a_{1}}+\frac{1}{a_{2}}+.......+\frac{1}{a_{n}})*(a_{1}+a_{2}+.....+a_{n}) \geq n^{2}$$ if and only if $$a_{1} =a_{2}=a_{3} .......=a_{n}$$

Proof :

Let P(n) : $$\frac{1}{a_{1}}+\frac{1}{a_2}+........+\frac{1}{a_{n}})(a_{1}+a_{2}+.......+a_{n}) \geq n^{2}$$ iff $$a_{1}= a_{2} = a_{3} =.......=a_{n}$$

Let n =

$$(\frac{1}{a_{1}}+\frac{1}{a_{2}}))(a_{1}+a_{2}) \geq 2^{2}$$