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Extra Credit

Module by: Kenny M. Felder. E-mail the author

Summary: An extra problem on compound interest to be used for extra credit. This module is part of the Teacher's Guide.

An extra cool problem you may want to use as an extra credit or something

Exercise 1

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?

Note:

(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.)

Solution

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+11+i1001+i100n+11+i100 size 12{ left [ left (1+ { {i} over {"100"} } right ) rSup { size 8{n+1} } - left (1+ { {i} over {"100"} } right ) right ]} {}

Example 1

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} } } {}[1.06311.06]=$419,008.39

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

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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