A bank gives
i
%
i% interest, compounded annually. (For instance, if
i
=
6
i=6, that means 6% interest.) You put
A
A dollars in the bank *every year for
n
n years*. At the end of that time, how much money do you have?

(The fine print: Let’s say you make your deposit on January 1 every year, and then you check your account on December 31 of the last year. So if
n
=
1
n=1, you put money in exactly once, and it grows for exactly one year.)

The money you put in the very *last* year receives interest exactly once. “Receiving interest” in a year always means being multiplied by
1+i1001+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}. (For instance, if you make 6% interest, your money multiplies by 1.06.) So the
A
A dollars that you put in the last year is worth, in the end,
A
1+i100A1+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}.

The *previous* year’s money receives interest twice, so it is worth
A
1+i1002A1+i1002 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{2} } } {} at the end. And so on, back to the first year, which is worth
A
1+i100nA1+i100n size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n} } } {} (since that initial contribution has received interest
n
n times).

So we have a Geometric series:

S
=
A
1+i100+
A
1+i1002+
...
+
A
1+i100nS=A1+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}+A1+i1002 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{2} } } {}+...+A1+i100n size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n} } } {}

We resolve it using the standard trick for such series: multiply the equation by the common ratio, and then subtract the two equations.

1+i100S
=
A
1+i1002+
...
+
A
1+i100n+
A
1+i100n+11+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}S=A1+i1002 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{2} } } {}+...+A1+i100n size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n} } } {}+A1+i100n+1 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n+1} } } {}

S
=
A
1+i100+
A
1+i1002
+
...
+
A
1+i100nS=A1+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}+A1+i1002 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{2} } } {}+...+A1+i100n size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n} } } {}

i100S
=
A
1+i100n+1–
A
1+i100i100 size 12{ left ( { {i} over {"100"} } right )} {}S=A1+i100n+1 size 12{ left (1+ { {i} over {"100"} } right ) rSup { size 8{n+1} } } {}–A1+i100 size 12{ left (1+ { {i} over {"100"} } right )} {}

S
=
100AiS=100Ai size 12{ { {"100"A} over {i} } } {}1+i100n+1−1+i1001+i100n+1−1+i100 size 12{ left [ left (1+ { {i} over {"100"} } right ) rSup { size 8{n+1} } - left (1+ { {i} over {"100"} } right ) right ]} {}

Example: If you invest $5,000 per year at 6% interest for 30 years, you end up with:

100(5000)6[
1.0631
–
1.06
]
=
$
419,008.39
100(5000)6 size 12{ { {"100" \( "5000" \) } over {6} } } {}[1.0631–1.06]=$419,008.39

Not bad for a total investment of $150,000!

Comments:"This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"