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Is there a better strategy to remedy the next while not having to jot down down each 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 annually. The speed of inflation and the financial institution rate of interest won’t ever change.
He’s about to obtain an inheritance and he desires to get simply sufficient cash that can final him precisely 30 years. He plans on depositing this quantity within the financial institution and get the 5% curiosity annually.
Adam has a yearly expense of 4402000 which he withdraws at first of every yr. His yearly bills will improve with inflation i.e his bills will improve by 6% annually. Assuming that Adam could have no different supply of revenue, what precise quantity does he want from his inheritance ?
Allow us to assume that Adam acquired y quantity as inheritance. Let x= 4402000. The standard strategy to remedy this might be the next :
Quantity Adam has on the finish of the first yr
= (y-x) * 1.05
Quantity Adam has on the finish of the 2nd yr
= ((y-x) * 1.05 – x * 1.06) * 1.05
= y * $1.05^2$ -x * $1.05^2$ -x * 1.06 * 1.05
Quantity Adam has on the finish of the third yr
= (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 * $1.05^{28}$ * $1.06^2$…..- $x*1.06^ {30}$
= 0
Now, I can write down each time period and remedy this. And it’ll not take an excessive amount of time both. However, if as a substitute of 30 years, we had been to calculate the inheritance quantity for , say, 1000 years then we might not have the option to take action. What’s a wiser manner of fixing this ?
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