csat-suneung 2009 Q23

csat-suneung · South-Korea · csat__math-humanities 4 marks Sequences and Series Recurrence Relations and Sequence Properties
Let $a _ { n }$ be 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 with equal quotient and remainder 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 $a _ { n }$ be 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 with equal quotient and remainder 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]