(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If you deposit $deposit into a bank account each year and get y% annual interest, the simple formula is

[tex]\frac{deposit((1+y)^{years}-1)}{y}[/tex]

How is this formula calculated? What's going on?

Let's look at $50, 6% interest, two years.

Maybe I'm supposed to be dropping $50 in just as the first year ends, so I don't build any interest on it the first year. I then build interest on it the second year, then drop in another $50 just as the year ends:

[tex]=(0*1.06+50)*1.06+50[/tex] no interest the first year + 50, then interest on that and another 50

=50*1.06+50

=103

That squares with the formula given earlier. So, let's look at three years

[tex]=(0*1.06+50)*1.06*1.06+50*1.06+50[/tex]

That could be rewritten as

[tex]=50x^2+50x+50[/tex] where x is 1+interest and the variable is (years-1, years-2... 1)

That comes to the same answer as the original formula, $159.18, so things are working out.

Let's look at 7 years. How does:

[tex]50*1.06^6+50*1.06^5+50*1.06^4+50*1.06^3+50*1.06^2+50*1.06+50[/tex]

get turned into:

[tex]\frac{50(1.06^7-1)}{.06}[/tex]

Let's play around with it.

[tex]50(1.06^6+1.06^5+1.06^4+1.06^3+1.06^2+1.06+1)[/tex]

So

[tex]1.06^6+1.06^5+1.06^4+1.06^3+1.06^2+1.06+1=\frac{1.06^7-1}{.06}[/tex]

right? Maybe if I explicitly state the interest...

[tex](1+.06)^6+(1+.06)^5+(1+.06)^4+(1+.06)^3+(1+.06)^2+(1+.06)+1[/tex]

[tex](1^6+.06^6)+(1^5+.06^5)+(1^4+.06^4)+(1^3+.06^3)+(1^2+.06^2)+(1+.06)+1[/tex]

[tex](1+.06^6)+(1+.06^5)+(1+.06^4)+(1+.06^3)+(1+.06^2)+(1+.06)+1[/tex]

[tex]1+.06^6+1+.06^5+1+.06^4+1+.06^3+1+.06^2+1+.06+1[/tex]

[tex].06^6+.06^5+.06^4+.06^3+.06^2+.06+7[/tex]

This isn't helping, it's not taking me in the direction that I want to go. How does:

[tex]50*1.06^6+50*1.06^5+50*1.06^4+50*1.06^3+50*1.06^2+50*1.06+50[/tex]

which is 419.6918825 get turned into:

[tex]\frac{50(1.06^7-1)}{.06}[/tex]

which is 419.6918825? How is that original equation calculated?

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# How to calculate the interest forumula

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