9. The U.S. Mint produces 7.4 billion pennies a year (see www.usmint.gov). Suppose
further that about 5% of the pennies are removed from circulation each year. (They
end up in peoples’ jars never to be seen again; I am using U.S. Mint’s figures: see
www.usmint.gov .) Assume that yearly production takes place at the start of any
given year and that the 5% are removed at the end of the year.

Start your analysis in some year past and tell me how many pennies you think are in circulation in U.S.
today?
[Hint: Start with production of 7.4 in any year (in your lifetime).
Then ask
what also remains in circulation from the production of the previous year?
What about from the year before that?
And so on… Go back forever; and then add all your terms up. This should generate an infinite geometric series for you.]

will pay more pixels as required

No math pros?

e/ oh lag double

This post was edited by Nearin on Oct 7 2014 08:57pm

just including this year:
7.4b * 0.95

this year + last:
((7.4b *0.95) + 7.4b) * 0.95

previous 3 years:

(((7.4b * 0.95) + 7.4b) * 0.95 + 7.4b) * 0.95

notice a pattern?

similarly, you should be able to do this in excel. tldr it approaches 140.6 billion

don8

This post was edited by carteblanche on Oct 7 2014 09:31pm

can't use Excel I need to find a formula to Do it representing it as Periods approach infinity

but I figured it out the answer was near 140 b It was 7.4b/0.05=148b

I was way over thinking it.

You can count the number of pennies in circulation at many moments during the year.

• If you count it at the start of the year, just after the new 7.4 billions : let Un be the number of pennies at this moment of year number n.
  Then Un+1 = 0.95 * Un + 7.4 billions
  
  Show that the sequence Vn = Un - 148 billions is a geometric sequence, of common ratio 0.95.
  Deduce the limit of Vn as n approaches infinity, then prove that the limit of Un is 148 billions. Notice that it doesn't depend on the initial number of pennies Uo.
  
• If you count it at the end of the year, just after the 5% reduction : let Wn be the number of pennies at this moment of year number n.
  Then Wn+1 = ( Wn + 7.4 billions ) * 0.95
  
  Show that the sequence Xn = Wn - 140.6 billions is a geometric sequence of common ratio 0.95.
  etc

Of course these 2 results are closely related : if there's 140.6 billions pennies at the end of the year, there's 140.6 + 7.4 = 148 billions at the start of next year.

Nerds jk I kinda figured it out while driving