A natural number k >1 is called good if there exist natural numbers a₁ < a₂ < < a_k such that 1 a₁ + 1 a₂ + + 1 a_k =1 Let f(n) be the sum of the first n good numbers, n 1 . Find the sum of all values of n for which f(n+5) / f(n) is an integer.

Step-by-step solution

Let a ₁= A ₁^2, a ₂= A ₂^2 a _ k = A _ k ^2 we have to check if it is possible for distinct natural number A ₁, ~A ₂ . _ _ _ _ A _ k to satisfy, 1 A₁ + 1 A₂ + . 1 A_k =1 For k=2 ; it is obvious that there do not exist distinct A₁, A₂ , such that 1 A₁ + 1 A₂ =1 2 is not a good number. For k =3 ; we have 1 2 + 1 3 + 1 6 =1 3 is a good number. 1 2 + 1 2 ( 1 2 + 1 3 + 1 6 )=1 4 is a good number. k will be a good numbers for all k 3 aligned & f(n)=3+4+ n terms = n(n+5) 2 ; f(n+5)= (n+5)(n+10) 2 & f(n+5) f(n) = n+10 n =1