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February 22nd, 2014, 03:11 PM   #1
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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 ti-83 (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.
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February 22nd, 2014, 04:29 PM   #2
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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.
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February 22nd, 2014, 07:37 PM   #3
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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
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February 23rd, 2014, 03:43 AM   #4
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It's known as "annuity due" rather than "annuity immediate".
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February 23rd, 2014, 07:12 AM   #5
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Re:

Quote:
Originally Posted by skipjack
It's known as "annuity due" rather than "annuity immediate".
From what I can see (googling), it's annuity due or immediate.
I'm sticking with immediate: due to me is confusing, so it's all yours :P
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February 23rd, 2014, 10:06 PM   #6
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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 interest-earning 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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