Let \(a_n\) be the amount of money he has in the account at the end of the \(n\)th month.
So at the end of the first month (assuming he doesn't make a withdrawal then) he would have
\(a_1=450\).
Halfway through the next month, he withdraws 1/3 of the account. The account doesn't earn interest, so only his withdrawals affect the amount of money in the bank. So at the end of the second month, he would have
\(a_2=\dfrac13(\text{current amount})=\dfrac13\times450\)
\(a_2=\dfrac13a_1\)
At the end of the third, \(a_3=\dfrac13(\text{new current amount})=\dfrac13a_2\)
And so on. So the recursive rule is
\(a_n=\dfrac13a_{n-1}\)
starting with \(a_1=450\).
