nn n f( x)=( a+ b-2 c) x^2+( b+ c-2 a) x+( c+ a-2 b)=0 n f( x)= a+ b-2 c+ b+ c-2 a+ c+ a-2 b=0 n f(1)=0 n Therefore alpha * 1=frac c+ a-2 b a+ b-2 c n alpha=frac c+ a-2 b a+ b-2 c n If,-1frac a+ c2nn nnnntherefore, b cannot be G.M. between a and c .nnnn n If, 0 c and b

For 0 < c < b < a, let (a + b - 2c)x^{2} + (b + c - 2a... | Sequence & Series (Mathematics)

\n\n\begin{aligned}\n& \mathrm{f}(\mathrm{x})=(\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}+(\mathrm{c}+\mathrm{a}-2 \mathrm{~b})=0 \\\n& \mathrm{f}(\mathrm{x})=\mathrm{a}+\mathrm{b}-2 \mathrm{c}+\mathrm{b}+\mathrm{c}-2 \mathrm{a}+\mathrm{c}+\mathrm{a}-2 \mathrm{~b}=0 \\\n& \mathrm{f}(1)=0 \\\n& \therefore \alpha \cdot 1=\frac{\mathrm{c}+\mathrm{a}-2 \mathrm{~b}}{\mathrm{a}+\mathrm{b}-2 \mathrm{c}} \\\n& \alpha=\frac{\mathrm{c}+\mathrm{a}-2 \mathrm{~b}}{\mathrm{a}+\mathrm{b}-2 \mathrm{c}} \\\n& \mathrm{If},-1<\alpha<0 \\\n& -1<\frac{\mathrm{c}+\mathrm{a}-2 \mathrm{~b}}{\mathrm{a}+\mathrm{b}-2 \mathrm{c}}<0 \\\n& \mathrm{~b}+\mathrm{c}<2 \mathrm{a} \text { and } \mathrm{b}>\frac{\mathrm{a}+\mathrm{c}}{2}\n\n\end{aligned}\n\n

\n\ntherefore, b cannot be G.M. between a and c .\n\n

\n\n\begin{aligned}\n& \text { If, } 0<\alpha<1 \\\n& 0<\frac{\mathrm{c}+\mathrm{a}-2 \mathrm{~b}}{\mathrm{a}+\mathrm{b}-2 \mathrm{c}}<1 \\\n& \mathrm{~b}>\mathrm{c} \text { and } \mathrm{b}<\frac{\mathrm{a}+\mathrm{c}}{2}\n\n\end{aligned}\n\n

\n \nTherefore,

\mathbf{b}

may be the G.M. between

\mathbf{a}

and

\mathbf{c}

.\n

Explanation

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