Find the sum of n terms of the series `(a+b)+(a^(2)+ab+b^(2))+(a^(3)+a^(2)b+ab^(2)+b^(3))+"......."` where `a ne 1,bne 1` and `a ne b`.

Explanation

` a ne1 b ne 0 and ane b `
Let S = ( a+ b) + `(a^(2) + ab + b^(2)+ ( a^(3) + a^(2) b+ ab^(2) +b^(3) + ….+ n ` terms
`=(1) /( (a-b) ) [(a^(2)-b^(2)) + ( a^(3) -b ^(3)) + (a^(4) -b^(4) ) + ....+_n terms ]`
`= (1) /((a-b) ) [a^(2) (1+ a+ ......+ nterms )- b ^(2)(1+ b+ b^(2) + .....+n terms ) `

` =(1)/( (a- b) ) [ a^(2)* (1* (a^(n) -1) )/( (a-1) ) -b^(2) ( 1* (b^(n) -1))/( (b-1) ) ]`
` " " = (1)/( (a-b ) ) [a^(2) ((1- n^(2)))/( (1- a) ) - b^(2) ( ( 1- b^(n)))/( (1-b) ) ]`