Let [tex]a_n[/tex] be the amount of money he has in the account at the end of the [tex]n[/tex]th month.
So at the end of the first month (assuming he doesn't make a withdrawal then) he would have
[tex]a_1=450[/tex].
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
[tex]a_2=\dfrac13(\text{current amount})=\dfrac13\times450[/tex]
[tex]a_2=\dfrac13a_1[/tex]
At the end of the third, [tex]a_3=\dfrac13(\text{new current amount})=\dfrac13a_2[/tex]
And so on. So the recursive rule is
[tex]a_n=\dfrac13a_{n-1}[/tex]
starting with [tex]a_1=450[/tex].
