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Algebra
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Just multiply and divide the expression by (a-b) and sum it.
The expression will become
S = [(a2-b2) + (a3-b3) + (a4-b4) +..........]/(a-b)
( Sum of infinite GP 1+a+a2+a3+...... = 1/(1-a) ----------- (Geometric Progressions formula) )
So a2+a3+a4+.... = a2/(1-a).
So S = [ (a2+a3+....) - (b2+b3+.....) ] / (a-b)
So S = [ a2/(1-a) - b2/(1-b) ] / (a-b)
or S = (a+b-ab)/[(1-a)(1-b)]
I DONT KNOW I AM RIGHT OR WRONG
LET
A= (a+b)+ (a^2 +ab+b^2) +(a^3+a^2b+ab^2+b^3)+..........................
A= (a+a^2+a^3+................)+(b+b^2+b^3+........)+(ab+ (a^2b+ab^2)+(a^3.b+a^2.b^2+a.b^3)+........)
A= (a+a^2+a^3+..............)+(b+b^2+b^3+.......)+(ab + ab.(a+b)+ ab.(a^2+ab+b^2)+..................... )
A= (a+a^2+a^3+............)+(b+b^2+b^3+......)+ab(1+ (a+b)+ (a^2+ab+b^2)+..........................)
A= (a+a^2+a^3+.........)+(b+b^2+b^3+.....)+ab(1+A)
A -ab.A=(a+a^2+a^3+...........)+(b+b^2+b^3+.................)+ab
A(1-ab)=(a+a^2+a^3+...........)+(b+b^2+b^3+................)+ab
as i know you are not fool you will never say that a and b are greater or equal to -1 or lesser or equal to -1
so
A(1-ab)= a/(1-a)+ b/(1-b) +ab
after this solve by yourself
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