Forum Notes Wednesday April 30, 2008
Lesson Opener 11.5
Answers
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Recursive Rules
--a different way to generate a sequence
--explicit rule ex: an=3n+7
--recursive rule ex: an= (an-1)2 +1
Generate the first 5 terms of the sequence.
a. a1= 1 an= (an-1)2 +1
Answer: 1,2,3,5,26,677
b. a1 = 2 a2 =2 an = an-2 - an-1
Answer: 2,2,0,2,-2
Factorial
Factorial symbol is !
n! represents the product of all the integers 1 to n
0! = 1 1! = 1
3! = 3*2*1= 6 4!= 4*3*2*1 = 24
Write an explicit and a recursive rule for a1 = 15 and d = 5
- explicit rule
an = 5n +10
- recursive rule
a1 = 15 an = an-1 +5
Write an explicit and a recursive rule for a1 = 4 and r = .2
- explicit rule
an = 4(.2)n-1
- recursive rule
a1 = 4 an = .2(an-1 )
Write a recursive rule for:
1,1,4,10,28,76
a1 = 1 a2 = 1 an = 2 (an-1 + an-2 )
A lake contains 5200 fish. The population of the fish declines 30% per year due to fishing and other causes. This lake is restocked with 400 fish each year.
- recursive rule
a1 = 5200 an = .7(an-1 ) +400
b) find the number of fish at the beginning of the 5th year
See Recursive Rules.doc on the OSP for how to do this on a calculator
About 1333 fish.
NOTE: You can find the “u” button on the calculator by pressing 2nd and 7
6000 trees are on a tree farm. Each year 25% of the trees are harvested and 900 new seedlings are planted.
a. write a recursive rule
a1 = 6000 an = .75(an-1 ) +9000
b. use the recursive rule to find the number of fish after 3 years.
4612 trees
c. graph the sequence over 10 years.
d. from the graph, what would you expect to happen with the number of trees in the tree farm?
You would expect the number of trees to level off and stop declining at 3600 trees.