fibonacci numbers, real numbers, golden
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please look at the files before doing it. thanks
12 pages
1) the defining properties of the fibonacci numbers, not in terms of rabbits but strictly in terms of the mathematical relationship among the terms.
2) back to rabbits. keep the assumptions of rabbit reproduction the same as for the fibonacci numbers, but with one change. now assume that the time to fertility for a newborn is 2 months rather than 1. starting with a newborn make and female, make a table which gives the number of pairs of rabbits after each month for 10 months. what is the mathematical relationship that gives the terms of this sequence of numbers?
3) now let's change the initial conditions. suppose that we have a sequence of numbers starting with 2,1 and the successor is as before the sum of its two predecessors. list the first 10 numbers of this sequence and the ratios of successive terms. mark mentioned in class that the limiting value for the rations of successive terms should be the golden ratio. find out if he is right or wrong by using the method from the worksheet that finds a quadratic equation solved by this limiting value. what changes? what stays the same?
4)construct an irrational number like 0.1011011101111.... and explain why it is irrational. don't use something obviously derived from this one, like 0.20220222...
12 pages
1) the defining properties of the fibonacci numbers, not in terms of rabbits but strictly in terms of the mathematical relationship among the terms.
2) back to rabbits. keep the assumptions of rabbit reproduction the same as for the fibonacci numbers, but with one change. now assume that the time to fertility for a newborn is 2 months rather than 1. starting with a newborn make and female, make a table which gives the number of pairs of rabbits after each month for 10 months. what is the mathematical relationship that gives the terms of this sequence of numbers?
3) now let's change the initial conditions. suppose that we have a sequence of numbers starting with 2,1 and the successor is as before the sum of its two predecessors. list the first 10 numbers of this sequence and the ratios of successive terms. mark mentioned in class that the limiting value for the rations of successive terms should be the golden ratio. find out if he is right or wrong by using the method from the worksheet that finds a quadratic equation solved by this limiting value. what changes? what stays the same?
4)construct an irrational number like 0.1011011101111.... and explain why it is irrational. don't use something obviously derived from this one, like 0.20220222...
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