Question 1202908
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You deposit $100 each month into an account earning 5% interest compounded monthly.
How much will you have in the account in 35 years?
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        This my solution goes second after the solution by @Theo.

        @Theo used an online calculator in his solution.

        Since this web-site is to teach Math, the expected solution should be mathematical.

        So, I present here a standard mathematical solution.

        Calculator can be used as a secondary tool for verification.

        Then the sequence of steps would be normally educative.



<pre>
I will assume that this account is an ordinary annuity saving plan, i.e. depositing and compounding
are made at the end of each month.


Use the formula for the Future Value of the ordinary annuity


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


where  FV is the future value of the account;  P is the monthly payment (deposit); 
r is the monthly effective compounding rate presented as a decimal; 
n is the number of deposits (= the number of years multiplied by 12, in this case).


Under the given conditions, P = $100;  r = 0.05/12;  n = 35*12 = 420 payments/(compounding).
So, according to the formula (1), you get for the monthly payment value


    P = {{{100*(((1+0.05/12)^(35*12)-1)/((0.05/12)))}}} = 113609.24  (rounded to the closest cent).


It is precisely the same value as @Theo obtained in his post using a calculator.


<U>Answer</U>.  The necessary monthly deposit value is $113609.24.
</pre>

Solved.



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In such calculations, it is important to avoid intermediate rounding.
Rounding can be done at the end, only.

Intermediate rounding is PROHIBITED, since it leads to wrong answer.


Also, a calculator should provide the necessary precision.


I use MS Excel in my computer. It works with 15 decimals in mantissa 
and provides the necessary precision.


I simply copy-paste my numerical formula into Excel spreadsheet and get the answer in one click.


Many standard online calculators provide the necessary precision.


You may use, for example, these free-of-charge online calculators


https://www.omnicalculator.com/finance/annuity-future-value


https://www.calculatorsoup.com/calculators/financial/future-value-annuity-calculator.php


https://www.calculator.net/future-value-calculator.html


They are reliable, have convenient user-friendly interface, have complete instructions and descriptions, 
provide the necessary precision and were checked million times. You may use them to check my/your calculations.



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There is another (= one more) reason why I produced and placed my solution here.


@Theo' posts used visual plots to support his solutions.
These plots were integral inseparable part of his solutions.
But some time ago, Theo left this forum and stopped supporting web-site with his plots.
As a result, you see now some colored spots in his posts, where his plots should be.
Due to this reason, @Theo's posts lost their educational meaning and value.
Therefore, I create my posts with my own mathematical solutions
to replace @Theo' solutions and provide meaningful mathematical content.