csat-suneung 2009 Q21

csat-suneung · South-Korea · csat__math-science 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence
Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]
Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]