I have a question that I would like to pose to a church youth group, but it involves math. I have a rudimentary understanding of how to figure out the problem (English teacher here!), but don't fully understand the math behind it. Here is the scenario: Suppose all 10 of you decided to convert 5 people to our faith, and committed to walking with these people for 5 years in order to help them grow. After 5 years, if each of you worked for 5 converts, there would be 50 new members of the faith. After those 5 years, suppose all of those 50 people took 5 people and over the course of 5 years led those people to the faith. And then those 250 people.... And so on and so on. After 50 years, how many people would be converted to the faith? How long would it take for the entire world to be converted to the faith? So I was wondering if you could tell me if these are questions high schoolers will be able to answer, and if they have trouble getting started, what is a good hint I can give them to get them started in the right direction? Thanks!
They could do it by straight arithmetic: set up a chart, figure out each year's total. If they choose to do it more mathematically, it's not a difficult problem-- if you know the formulas. (Our kids see it Junior Year.) It's called a Geometric Sequence, and they want to find the General Term. In this particular case, a1 (the first number) is 50, n=10 (since there are 10 increments to get you to that 50 year mark-- the fact that each one takes 5 years is immaterial as far as the math goes) and r (the number you're multiplying by each time) is 5. It's going to be a BIG number... warn them that a typical scientific calculator may have to round it off using scientific notation. Sorry, I don't have my calculator here and I'm in the middle of schoolwork.
Oops, I realized I never gave you the formula--- sorry: x = a1*r ^(n-1) or 50(5^10 - 1) or 50( 5 ^ 9) Remind them to raise 5 to the 9th BEFORE they multiply it by 50-- its' called Order of Operations.
Am I missing something...Don't we need to add them all up after the first 5 years we have 50 new people converted then those 50 people convert 250 new people...so at this point we have 50+250=300 people converted since the beginning of this process...and so on.
I had a long answer posted here, using the geometric series formula...think I might need geometric sequence...with series, I got S(sub 11) = 122,070,310, then thought I should subtract out the original 10 people, so that would make 122,070,300. But should I have used the geometric sequence formula instead? (Edited to fix division error!)
Inexcusably careless mistake... sorry about that. The correct formula is: [a1(1-r^n)] / (1-r) It's the formula for the sum of the terms of a geometric sequence. I get 122,070,300. Now, given my litany of excuses- still recovering from last weekend, the insanity that IS NHS for graduation Sunday, would someone please check my math??
