Use mathematical induction to prove the following.If a is a constant and 0 < a < 1, then an < an-1.
What will be an ideal response?
Answers may vary. One possibility:
Sn: If a is a constant and 0 < a < 1, then an < an-1.
S1: If a is a constant and 0 < a < 1, then a1 < a1-1.
Sk: If a is a constant and 0 < a < 1, then ak < ak-1.
Sk+1: If a is a constant and 0 < a < 1, then ak+1 < ak.
1. Basis step: Since a1-1 = a0 = 1 and it is given that a < 1 (which implies a1 < 1), S1 is true.
2. Induction step: Let k be any natural number. Assume Sk. Deduce Sk+1.
If a is a constant and 0 < a < 1, then ak < ak-1.
If a is a constant and 0 < a < 1, then ak ? a < ak-1 ? a. Multiplying by a, a > 0
If a is a constant and 0 < a < 1, then ak+1 < a(k-1)+1.
If a is a constant and 0 < a < 1, then ak+1 < ak.
You might also like to view...
Find the domain of the expression.
?
?
A. all real numbers B. all real numbers except x =-7 C. all real numbers except x = 8 D. all real numbers except x = 8 and x = -7 E. x = 8 or x = -7
Solve the equation for exact solutions over the interval [0, 2?).2 sin2x = sin x
A. 
B. 
C. 
D. 
Graph the pair of parametric equations in the rectangular coordinate system.x = t2, y = 
+ 4; 0 ? t ? 4 
A. 
B. 
C. 
D. 
A group of fun-loving people have decided to play a practical joke on one of their friends, but they can't decide which friend, Alice (A), Betty (B), or Connie (C). Their preferences are: (ABC) (CBA) (BCA) 13 4 7 ? Who wins using the Borda count method? Does this violate the Condorcet criterion? ?
A. Betty, No B. Connie, No C. Alice, Yes D. Betty, Yes E. Alice, No