Let S be the infinite sum given by S = _ n =0 ^ a _ n 10^ 2 n where a_ n _ n 0 is a sequence defined by a₀=a₁=1 and a_ j =20 a_ j-1 -108 a_ j-2 for j 2 . If S is expressed in the form a b , where a, b are coprime positive integers, then a equals

Step-by-step solution

array l a_ n =20 a_ n-1 -108 a_ n-2 a_ n 10^ 2 n = 20 a_ n-1 10^ 2 n - 108 a_ n-2 10^ 2 n a_ n 10^ 2 n = 20 100 a_ n-1 10^ 2(n-1) - 108 10000 a_ n-2 10^ 2(n-2) array apply summation array l _ n =2 ^ a _ n 10^ 2 n = 1 5 _ n =2 ^ a _ n -1 10^ 2( n -1) - 27 2500 _ n =2 ^ a _ n -2 10^ 2( n -2) S -1- 1 100 = 1 5 ( ~S -1)- 27 2500 ~S ~S -1- 1 100 = 1 5 ~S - 1 5 - 27 2500 ~S ~S (1- 1 5 + 27 2500 )=- 1 5 +1+ 1 100 array ~S ( 2500-500+27 2500 )= 81 100 ~S = 81 25 2027 ~S = 2025 2027