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Question

Find the sum of the following geometric series:
(x+y)+(x2+xy+y2)+(x3+x2y+xy2+y3)+ to n terms;

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Solution

Let Sn=(x+y)+(x2+xy+y2)+(x3+x2y+xy2+y3)+n terms;
Multiply and divide with (x-y) then we obtain
Sn=(xy)[(x+y)+(x2+xy+y2)+(x3+x2y+xy2+y3)+ n terms(xy)
Sn=[(xy)(x+y)+(xy)(x2+xy+y2)+(xy)(x3+x2y+xy2+y3)+n terms](xy)
Sn=[(x2y2)+(x3y3)+(x4y4)+n terms](xy)
Sn=(x2+x3+x4+n terms)y2+y3+y4n terms(xy)
Sn=[x2(1+x+x2+nterms)y2(1+y+y2+n terms)](xy)
Sn={x2(1xn)1x}{y2(1yn)(1y)}(xy)



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