Is there a better strategy to clear up the next without having to write down down every time period ?

Adam lives in a rustic the place there isn’t any revenue tax. The annual price of inflation on this nation is 6%. The banks of this nation give 5% curiosity every year. The price of inflation and the financial institution rate of interest won’t ever change.

He is about to obtain an inheritance and he desires to get simply sufficient cash that may final him precisely 30 years. He plans on depositing this quantity within the financial institution and get the 5% curiosity every year.

Adam has a yearly expense of 4402000 which he withdraws initially of every 12 months. His yearly bills will enhance with inflation i.e his bills will enhance by 6% every year. Assuming that Adam may have no different supply of revenue, what precise quantity does he want from his inheritance ?

Let us assume that Adam obtained y quantity as inheritance. Let x= 4402000. The conventional strategy to clear up this is able to be the next :

Amount Adam has on the finish of the first 12 months

= (y-x) * 1.05

Amount Adam has on the finish of the 2nd 12 months

= ((y-x) * 1.05 – x * 1.06) * 1.05

= y * $1.05^2$ -x * $1.05^2$ -x * 1.06 * 1.05

Amount Adam has on the finish of the third 12 months

= (y * $1.05^2$ -x * $1.05^2$ -x * 1.06 * 1.05 – x * $1.06^2$) * 1.05

= (y * $1.05^3$ -x * $1.05^3$ -x * $1.05^2$ * 1.06- x * $1.06^2$ * 1.05)

Extrapolating the above sample, after 30 years , the equation turns into

y * $1.05^{30}$– x * $1.05^{30}$ – x * $1.05^{29}$ * 1.06 – x * $1.05^{28}$ * $1.06^2$…..- $x*1.06^ {29}$ * 1.05 = 0

Now, I can write down every time period and clear up this. And it is not going to take an excessive amount of time both. But, if as an alternative of 30 years, we have been to calculate the inheritance quantity for , say, 1000 years then we might not find a way to take action. What is a wiser means of fixing this ?