If a, b, c are in A.P. and a^2, b^2, c^2 are | vedclass

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(A) Given that $a, b, c$ are in $A.P.$$2b = a + c$ ......$(i)$Given that $a^2, b^2, c^2$ are in $H.P.$$b^2 = \frac{2a^2c^2}{a^2 + c^2}$$b^2(a^2 + c^2)

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$a = b = c$

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(A) Given that $a, b, c$ are in $A.P.$$2b = a + c$ ......$(i)$Given that $a^2, b^2, c^2$ are in $H.P.$$b^2 = \frac{2a^2c^2}{a^2 + c^2}$$b^2(a^2 + c^2) = 2a^2c^2$$b^2((a+c)^2 - 2ac) = 2a^2c^2$Substituting $a+c = 2b$ from $(i)$:$b^2(4b^2 - 2ac) = 2a^2c^2$$4b^4 - 2acb^2 - 2a^2c^2 = 0$$2b^4 - acb^2 - a^2c^2 = 0$$(2b^2 + ac)(b^2 - ac) = 0$Since $a, b, c$ are real,$2b^2 + ac = 0$ implies $a, b, c$ are $0$ or complex,so we consider $b^2 = ac$.If $b^2 = ac$,then $a, b, c$ are in $G.P.$Since $a, b, c$ are in both $A.P.$ and $G.P.$,we must have $a = b = c$.

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $H.P.$,then

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