Let 
be the amount of money he has in the account at the end of the 
th month.
So at the end of the first month (assuming he doesn't make a withdrawal then) he would have
.
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
At the end of the third,
And so on. So the recursive rule is
starting with.
So at the end of the first month (assuming he doesn't make a withdrawal then) he would have
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
At the end of the third,
And so on. So the recursive rule is
starting with