If 1+ 3 - 2 2 3 + 5-2 6 18 + 9 3 -11 2 36 3 + 49-20 6 180 + upto =2+ ( b a +1 ) _e ( a b ) , where a and b are integers with gcd ( a , b )=1 , then 11 a +18 ~b is equal to ______

Step-by-step solution

aligned & S=1+ x 2 3 + x^2 18 + x^3 36 3 + x^4 180 + & Put x 3 =t, where x= 3 - 2 & S=1+ t 2 + t^2 6 + t^3 12 + t^4 20 + & S=1+t (1- 1 2 )+t^2 ( 1 2 - 1 3 )+t^3 ( 1 3 - 1 4 )+t^4 ( 1 4 - 1 5 ) & S= (1+t+ t^2 2 + t^3 3 + t^3 4 + )- ( t 2 + t^2 3 + t^3 4 + t^4 5 + ) & S= (t+ t^2 2 + )- 1 t (t+ t^2 2 + t^3 3 + )+2 & S=2+ (1- 1 t )(- (1-t))= ( 1 t -1 ) (1-t)+2 & S=2+ ( 3 3 - 2 -1 ) (1- 3 - 2 3 ) & S=2+ ( 2 3 - 2 ) e 2 3 & S=2+ ( 6 +2) 2 e 2 3 =2+ ( 3 2 +1 ) e 2 3 & a=2, b=3 & 11 a+18 b=11 2+18 3=76 aligned