# Suppose you are proving that an - Expert Answers

Question 15
Suppose you are proving that an integer derived in a certain way is even, and that you are at a point in the proof that you have the integer expressed as 7(4k2 + 4k + 1) + 1. How should you rewrite this expression to show that the integer is even?
28k2 + 28k + 7 + 1
28k2 + 28k + 8
20(k2 + k) + 8(k2 + k + 1)
2(14k2 + 14k + 4)
Question 16
Which one of the following gives a recursive algorithm for computing an, where n is a positive integer and a is a real number?
procedure power(a: real number, n: positive integer)
if n = 1 then power(a, n) := a
else power(a, n) := a∙power(a, n)
procedure power(a: real number, n: positive integer)
for i = 1 to n
power := a∙power
procedure power(a: real number, n: positive integer)
power := a
for i = 2 to n
power := a∙power
procedure power(a: real number, n: positive integer)
if n = 1 then power(a, n) := a
else power(a, n) := a∙power(a, n – 1)
Question 17
Let f be the function such that f(0) = 1, f(1) = 4, and
f(n) = f(n – 1)∙f(n – 2) + 1 for all integers n greater than 1. Which one of the following is true?
f(3) = 4 and f(4) = 105
f(3) = 5 and f(4) = 21
f(3) = 21 and f(4) = 106
f(3) = 20 and f(4) = 105
Question 18
If {an} is the sequence defined by an = (n + 5)/2 for all positive integers n, which one of the following is a recursive definition for the sequence?
a1 = 1 and, for n = 2, 3, 4, . . . ,
an = an–1 + 5/2
a1 = 1 and, for n = 2, 3, 4, . . . ,
an = an–1 + 1/2
a1 = 3 and, for n = 2, 3, 4, . . . ,
an = an–1 + 5/2
a1 = 3 and, for n = 2, 3, 4, . . . ,
an = an-1 + 1/2
Question 19
Use the graph below to match each quantity with the correct value.:
total number of vertices?
number of edges connecting a and c?
number of edges connecting a and b?
degree of vertex e?