Question 172901
value of the house today is 500,000
in 5 years, the house will appreciate 5 years worth of interest which makes the formula for the future value of the house:
500,000 * (1.05)^5 = 638140.78
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if he invests in a project at the end of each year and the project earns 12% a year, then he would have to invest 100449.57 each year to equal 638140.78 at the end of the 5th year.
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this is how it works out:
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at the end of the first year he invests: $100,449.57
at the end of the second year this is multiplied by 1.12 to equal: $112,503.52 
he invests an additional $100,449.57 to make this equal: $212,953.09
at the end of the third year this is multiplied by 1.12 to equal: $238,507.46
he invests an additional $100,449.57 to make this equal: $338,957.03
at the end of the fourth year this is multiplied by 1.12 to equal: $379,631.87
he invests an additional $100,449.57 to make this equal: $480,081.44
at the end of the fifth year this is multiplied by 1.12 to equal: $537,691.22
he invests an additional $100,449.57 to make this equal: $638,140.79
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he has invested 100,449.57 at the end of each year and this has accumulated and grown to equal $638,140.79 by the end of the fifth year.
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the value of the house has grown 5% per year for 5 years to equal $638,140.79.
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he has invested just enough money to buy the house with cash assuming no taxes have to be paid on the investment.
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there is a formula to derive this.
that formula has to do with the annuity from a future amount.
i used a calculator to get the answer.
if you had to use the formula, the formula would be:
{{{PMT = (FV/((((1+i)^n-1)/i)))}}}
which is the same as:
{{{PMT = (FV*i)/(((1+i)^n-1))}}}
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FV is the future value which is the same as future amount.
i is the periodic interest rate which is annual in this case.
n is the number of time periods which is years in this case.
PMT is the annual payment.
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this formula is for a payment at the end of a period which was what your problem was asking about.
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i checked the formula out and it works.