For positive integers n , if 4 a_n= (n^2+5 n+6 ) and S_n= _ k=1 ^n ( 1 a_k ) , then the value of 507 S₂₀₂₅ is :

Step-by-step solution

aligned & S_n= _ k=1 ^n 4 K^2+5 k+6 & = _ k=1 ^n 4 (K+2)(K+3) =4 _ K=1 ^n ( 1 K+2 - 1 K+3 ) aligned aligned & =4 [ 1 3 - 1 4 ] & =4 [ 1 4 - 1 5 ] & =4 [ 1 n+2 - 1 n+3 ] & S_n=4 [ 1 3 - 1 n+3 ] aligned aligned & S₂₀₂₅=4 [ 1 3 - 1 2028 ] & S₂₀₂₅=4 [ 675 2028 ] & 507 S₂₀₂₅=675 aligned