If `a, b, c` are in `G. P.`, prove that `a^2+ b^2, ab + bc, b^2 +c^2` are also in `G.P`

Explanation

Given, `a, b, c` are in `G. P.`


Geometric mean relation: `b^2=ac`


To prove: `a^2+ b^2, ab + bc, b^2 +c^2` are also in `G.P`


To prove: `(ab+bc)^2=(b^2 +c^2)(a^2+ b^2)`


`RHS=(b^2 +c^2)(a^2+ b^2)`


`=a^2b^2+a^2c^2+b^4+b^2c^2`


`=a^2b^2+a^2c^2+b^4+b^2c^2`


`=a^2b^2+a^2c^2+a^2c^2+b^2c^2`


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