If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x^{2}, b^{2}, y^{2} are in A.P.

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#### Solution

\[\text { a, b and c are in A . P } . \]

\[ \therefore 2b = a + c . . . . . . . (i)\]

\[\text { a, x and b are in G . P } . \]

\[ \therefore x^2 = ab . . . . . . . (ii)\]

\[\text { And, b, y and c are also in G . P } . \]

\[ \therefore y^2 = bc . . . . . . . (iii)\]

\[\text { Now, putting the values of a and c: } \]

\[ \Rightarrow 2b = \frac{x^2}{b} + \frac{y^2}{b}\]

\[ \Rightarrow 2 b^2 = x^2 + y^2 \]

\[\text { Therefore,} x^2 , b^2 \text { and } y^2 \text { are also in A . P } . \]

Concept: Geometric Progression (G. P.)

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