By Bradley Jackson, Dmitri Thoro
Publication via Jackson, Bradley, Thoro, Dmitri
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Extra info for Applied Combinatorics With Problem Solving
And prove that it holds for all n 1. 12. A ball is dropped from a height of 4 feet, and each time it hits the ground it rebounds to 3/4 the previous height. What is the total distance the ball will have traveled (up and down) when it reaches the top of its tenth rebound? 13. At the beginning of each year, 100 dollars is deposited into a savings account. At the end of each year, 5% interest is paid on all the money that was in the account at the beginning of the year. Give a formula for the amount of money in the account at the beginning of year n.
B) Write a program that finds, for a given positive integer n, the number of terms in the Fibonacci sequence that are less than or equal to n. c) Let a and b be given as in Problem 18(b) of this section. There we obtained the Binet form of = — b). Use a computer or calculator to discover a similar formula for Hint: Work with and Can you prove that this formula always works? d) For small values of n it appears that = [(a"/(a — b)) + 1/2]. ) Use a computer to investigate this conjecture for all n 25.
The following theorem shows that it is always possible to represent an integer using any base b, where b > 1. THEOREM I If b is any integer greater than 1, every integer a> 0 can be uniquely represented in the form a = 0, + + ... ,n. Proof For any positive integer a < b it is easy to represent a using base b, by taking n= 0 and r0 = a, and this representation is obviously unique. Now suppose that a b and let us assume that every positive integer less than a has a unique representation using base b.
Applied Combinatorics With Problem Solving by Bradley Jackson, Dmitri Thoro