Supplement to the Classical Algebra text

Chapter 3: Induction and the Binomial Theorem

PROBLEM SET 3 - Extra Problems

    78 - 82. Find the value of each recursive mystery function on any n-tuple  ( x1, x2, ..., xn )  and prove that your value is correct.

78.
myst ( x1, x2, ..., xn ) = x1 if n = 1
xn - myst ( x1, ..., xn-1 ) if n > 1
79.
myst ( x1, x2, ..., xn ) = x1 if n = 1
xn myst ( x1, ..., xn-1 ) if n > 1
80.
myst ( x1, x2, ..., xn ) = x1 if n = 1
xn if xn > myst ( x1, x2, ..., xn-1 )
myst ( x1, x2, ..., xn-1 ) otherwise
81.
myst ( x1, x2, ..., xn ) = x1 if n = 1
x1 - 2 myst ( x2, ..., xn ) if n > 1
82.
myst ( x1, x2, ..., xn ) = x1 if n = 1
myst ( x1, ..., xn-1 ) + myst ( x2, ..., xn ) if n > 1

 

 83. Find a recursive definition for the function

1 1 1
e ( n ) = 1 + - + - + ... + -
1! 2! n!
that gives an approximation to e, the base of natural logarithms.

 84. Discuss the following recursive definition of the GCD for a >= 0 and b >=0. Is it correct? Is it efficient?

GCD ( a, b ) = b if a = 0
GCD ( b, a ) if b < a
GCD ( b - a, a ) if b >= a > 0
© 1998 William J. Gilbert and Scott A. Vanstone, University of Waterloo

Supplement Contents   |   Section 3.2