Let a_ n be a sequence such that a₀=0, a₁= 1 2 and 2 a_ n +2 =5 a_ n +1 -3 a_ n , n =0,1,2,3, . Then _ k =1 ¹⁰⁰ a_k is equal to

Step-by-step solution

aligned & a₀=0, a₁= 1 2 & 2 a_ n+2 =5 a_ n+1 -3 a_n & 2 x^2-5 x+3=0 x=1,3 / 2 & a_n=A l^n+B ( 3 2 )^n aligned . array ll n =0 & 0= A + B n =1 & 1 2 = A + 3 2 ~B array ] aligned & A =-1 & ~B =1 aligned aligned & a _ n =-1+ ( 3 2 )^ n & _ k =1 ¹⁰⁰ a _ k = _ k =1 ¹⁰⁰(-1)+ ( 3 2 )^ k aligned aligned & =-100+ ( 3 2 ) ( ( 3 2 )¹⁰⁰-1 ) 3 2 -1 & =-100+3 ( ( 3 2 )¹⁰⁰-1 ) & =3 ( a ₁₀₀ )-100 aligned