(x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2} y + xy^{2} + y^{3}) + …. to n terms;

Let S_{n} = (x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2} y + xy^{2} + y^{3}) + …. to n terms

Let us multiply and divide by (x – y) we get,

S_{n} = \(\dfrac{ 1}{(x – y) [(x + y) (x – y) + (x^2 + xy + y^2) (x – y)}\) … upto n terms]

(x – y) S_{n} = (x^{2} – y^{2}) + x^{3} + x^{2}y + xy^{2} – x^{2}y – xy^{2} – y^{3}..upto n terms

(x – y) S_{n =} (x^{2} + x^{3} + x^{4}+…n terms) – (y^{2} + y^{3} + y^{4} +…n terms)

By using the formula,

Sum of GP for n terms = \(\dfrac{ a(1 – r^n )}{(1 – r)}\)

We have two G.Ps in above sum, so,

S_{n} = \( \dfrac{1}{(x-y) {x^2 [(x^n – 1)}{ (x – 1)] – y^2 [(y^n – 1)} (y – 1)]}\)

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 \( \dfrac{81}{2}\).