1. Prove that 1-11+2-21+ whenever " isa positive integer.
2. Give recursive definition of the sequence [a] n=1,2,3,
c) -nun+ 1)
3. Give recursive algorithm for finding the maximum of finite set of integers, making use of
the fact that the maximum of nintegersis the larger of thelast integer in the list and the
maximum of the first n-1 integers in the list.
4. Aparticular brand of shirt comes in 12 colors, has male version and female version and
comes in three sizes for each sex. How many different types of this shirt are made?
5 How many positive integersless than 1000
a) are divisible by 7?
b) are divisible by 7but not by 11?
c) are divisible by both and 11?
d) are divisible by either 7or 11?
e)are divisible by exactly one of and 11?
f) are divisible by neither nor 11?
g) have distinct digits?
h) have distinct digits and are even?
6. There are six different candidates for governor of state. In how many orderscan the names
of the candidates be printed on ballot?
7. Describe graph model that represents subway system in large city. Should edges be
directed or undirected? Should multiple edges be allowed? Should loopsbe allowed?
8. Represent this graph with an adjacency matrix.
Does each of these lists of vertices forma path in the following graph? Whichpaths are
simple? Which are circuits? What are the lengths of those that are paths?
a) a,b, e, c,b
c) e, a
10. What is the value of each of these postfix expressions?
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