Question 1198670
.
Mr. Gordon plans to invest $300 at the end of each month in an account that pays 6%, 
compounded monthly. After how many months will the account be worth $30,000?
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<pre>
The formula for an Ordinary Annuity saving account compounded monthly  is


    FV = {{{P*(((1+r/12)^n-1)/((r/12)))}}}


where FV is the future value, P is the monthly payment at the end of each month, 
r is the interest rate per year expressed as a decimal, 
n is the number of monthly deposits (the number of months).


So, we need to find " n " from this equation


    {{{((1+0.06/12)^n-1)/((0.06/12))}}} = {{{FV/P}}} = {{{30000/300}}} = 100,  


Thus the equation is

    {{{((1+0.005)^n-1)/0.005}}} = 100.


Rewrite it in this form

    {{{1.005^n-1}}} = 0.005*100,

    {{{(1.005)^n}}} = 1 + 0.005*100 = 1.5.


Take the logarithm base 10 of both sides

    n*log(1.005) = log(1.5)


and calculate  

     n = {{{log((1.5))/log((1.005))}}} = 81.296  months.


We should round it to the closest greater integer, which is 82 months, 
in order for the bank be in position to make the last compounding.
    

<U>ANSWER</U>. 82 months.


<U>CHECK</U>.  {{{300*((1.005^82-1)/0.005)}}} = 30316.75, which is greater than 30000;

        {{{300*((1.005^81-1)/0.005)}}} = 29867.41, which is less than 30000.
</pre>

Solved, checked, explained and completed.


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On ordinary annuity saving plan, &nbsp;see my lessons in this site 

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