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February 22nd, 2014, 03:11 PM  #1 
Member Joined: Dec 2013 Posts: 31 Thanks: 0  Calculate interest
I came up with this problem today and unable to think of a solution(without programming ): How to calculate the interest of regular increase of principal? EXAMPLE: Someone put USD 3000 into his bank account every year, which has a 5% interest rate and compounds once a year. He never spend money in this account. How much will he get in his account after 20 years of doing this. (on the starting day he'll put the first 3000 in the new account) I know it's probably easy to do it on a programable calculator like ti83 (don't have it handy) If I would have to do this problem, like on a test or so, I would do it like this: P=A(1+0.05)^t where A=money put in, P=money after certain amount of time(t)(ignore it if you don't like the distribution of alphabets for variables) Then P'=3000*1.05^20+3000*1.05^19+......+3000*1.05^2+30 00*1.05 The calculation is not hard to do with a calculator, but I don't really like this solution and think there must be something more beautiful in terms of mathematics. 
February 22nd, 2014, 04:29 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,927 Thanks: 2205 
1.05P' = 3000*1.05^21+3000*1.05^20+......+3000*1.05^3+3000* 1.05^2 = P' + 3000*1.05^21  3000*1.05, so P' = (3000*1.05^21  3000*1.05)/(1.05  1) = 60000(1.05^21  1.05) = 63000(1.05^20  1). The total interest is then given (in USD) by P'  60000. 
February 22nd, 2014, 07:37 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: Calculate interest
As a formula (known as annuity immediate): F = D(1+i)[(1+i)^n  1] / i F = Future value (?) D = periodic Deposit n = number of periods i = (interest rate)/100 
February 23rd, 2014, 03:43 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,927 Thanks: 2205 
It's known as "annuity due" rather than "annuity immediate".

February 23rd, 2014, 07:12 AM  #5  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: Quote:
I'm sticking with immediate: due to me is confusing, so it's all yours :P  
February 23rd, 2014, 10:06 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,927 Thanks: 2205 
The term "annuity immediate" or "ordinary annuity" is used when the payments are made at the end of each period, whereas "annuity due" is used when the payments are made at the beginning of each period, as is normally the case for an interestearning bank account. This is explained here and in this article. When a single letter is used for the future value (and payments are made at the beginning of each period), it's usually the letter S rather than F.


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