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

So, we have

bc = ad … (1)

b^{2} = ac … (2)

c^{2} = bd … (3)

Taking the R.H.S. we have

R.H.S.

= (ab + bc + cd)^{2}

= (ab + ad + cd)^{2} [Using (1)]

= [ab + d (a + c)]^{2}

= a^{2}b^{2} + 2abd (a + c) + d^{2} (a + c)^{2}

= a^{2}b^{2} +2a^{2}bd + 2acbd + d^{2}(a^{2} + 2ac + c^{2})

= a^{2}b^{2} + 2a^{2}c^{2} + 2b^{2}c^{2} + d^{2}a^{2} + 2d^{2}b^{2} + d^{2}c^{2} [Using (1) and (2)]

= a^{2}b^{2} + a^{2}c^{2} + a^{2}c^{2} + b^{2}c^{2 }+ b^{2}c^{2} + d^{2}a^{2} + d^{2}b^{2} + d^{2}b^{2} + d^{2}c^{2}

= a^{2}b^{2} + a^{2}c^{2} + a^{2}d^{2 }+ b^{2 }× b^{2} + b^{2}c^{2} + b^{2}d^{2} + c^{2}b^{2} + c^{2 }× c^{2} + c^{2}d^{2}

[Using (2) and (3) and rearranging terms]

= a^{2}(b^{2} + c^{2} + d^{2}) + b^{2} (b^{2} + c^{2} + d^{2}) + c^{2} (b^{2}+ c^{2} + d^{2})

= (a^{2} + b^{2} + c^{2}) (b^{2} + c^{2} + d^{2})

= L.H.S.

Thus, L.H.S. = R.H.S.

Therefore,

(a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d^{2}) = (ab + bc + cd)^{2}

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Find the geometric means of the following pairs of numbers:

**(i) **2 and 8

**(ii) **a^{3}b and ab^{3}

**(iii) **–8 and –2

Insert 5 geometric means between \( \dfrac{32}{9}\) and \( \dfrac{81}{2}\).