_ k=1 ⁴⁰ ( 1+ 1 k^2 + 1 (k+1)^2 )=a+ b c where a, b, c N , b < c, gcd (b, c)=1 , then what is the value of a+b ?

Step-by-step solution

Let aligned t_k & = 1+ 1 k^2 + 1 (k+1)^2 & = k^2(k+1)^2+k^2+(k+1)^2 k^2(k+1)^2 & = k^4+2 k^3+3 k^2+2 k+1 k^2(k+1)^2 = (k^2+k+1 )^2 k^2(k+1)^2 & = k^2+k+1 k(k+1) =1+ 1 k(k+1) & =1+ 1 k - 1 k+1 aligned aligned & S_n= _ k=1 ^n t_k=t₁+t₂+ +t_n & =n+1- 1 n+1 =(n+1)- 1 n+1 =n+ n n+1 & _ k=1 ⁴⁰ 1+ 1 k^2 + 1 (k+1)^2 =40+ 40 41 & a=b=40, c=41 & a+b=80 . aligned