Calculate high interest rate accounts

Re: Calculate high interest rate accounts


You're asking two questions--the first involves the $5K in the bank for 25 years--that is the easier to understand.

At the end of the first year you've got the principal ($5K) plus the interest [$5Kx(0.03)]. This can be expressed as
A1=$5K+0.03x$5K
using the distributive rule this becomes
A1=$5K(1+0.03) =$5kx1.03

Generally if the interest rate is i (decimal) the initial amount (A0) at the end of the first year is
A1=A0(1+i)

Using a similar development the amount at the end of the seasond year is

A2=A1+A1*i=(A0+A0*i)+(A0+A0*i)*i=A0+2A0*i+A0*i^2
A2=A0(1+2i+i^2)=A0(1+i)^2

A similar development for n years is similarly shown to be

An=A0(1+i)^n

So if A0=$5K and i=0.03 after 25 years that $5K has turned into
A25=$5K(1.03)^25

The second part is more difficult, but is based upon the same operation as the first, its just that it has been sweetened by your savings. Lets look at this a year at a time instead of daily

At the end of the first year you've added $720 to the $5k (plus interest) you've deposited so now you've got
A1=5000(1.03)+720

at the end of the second year you've got
A2=5000(1.03)^2+720(1.03)+720

third year
A3=5000(1.03)^3+720(1.03)^2+720(1.03)+720

nth year
An=5000(1.03)^n+720(1.03)^(n-1)+720(1.03)^(n-2)+....+720
using the distributive law this becomes
An=5000(1.03)^n+720[(1.03)^(n-1)+(1.03)^(n-2)+....+(1.03)^0]

There's a simpler expression for this messy series
it's

[(1+i)^n-1]/i

where i is interest (decimal) and n is the number of periods invested

BTW--I'm not an accountant--but I think this is normally called an annuity.

Rap