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If a, b, c are in A.P. and a, x, b, y, c...

If a, b, c are in A.P. and a, x, b, y, c are in G.P., then prove that `b^(2)` is the arithmatic mean of `x^(2)" and "y^(2).`

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To solve the problem, we need to prove that \( b^2 \) is the arithmetic mean of \( x^2 \) and \( y^2 \) given that \( a, b, c \) are in Arithmetic Progression (A.P.) and \( a, x, b, y, c \) are in Geometric Progression (G.P.). ### Step-by-Step Solution: 1. **Understanding A.P. Condition**: Since \( a, b, c \) are in A.P., we have: \[ b - a = c - b ...
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a, b, x are in A.P., a,b,y are in G.P. and a, b, z are in H.P. then: