( ◦ )() + ( ◦ )() = sin + sin = 2sin. The two sides are not equal, so the given statement is false.
19. Let be the statement that 7 − 1 is divisible by 6
• 1 is true because 7 1 − 1 = 6 is divisible by 6.
• Assume is true, that is, 7 − 1 is divisible by 6. In other words, 7 − 1 = 6 for some positive integer . Then
7 +1 − 1 = 7 · 7 − 1 = (6 + 1) · 7 − 1 = 42 + 6 = 6(7 + 1), which is divisible by 6, so +1 is true.
• Therefore, by mathematical induction, 7 − 1 is divisible by 6 for every positive integer .
20. Let be the statement that 1 + 3 + 5 + ··· + (2 − 1) = 2 .
• 1 is true because [2(1) − 1] = 1 = 1 2 .
• Assume is true, that is, 1 + 3 + 5 + ··· + (2 − 1) = 2 . Then
1 + 3 + 5 + ··· + (2 − 1) + [2( + 1) − 1] = 1 + 3 + 5 + ··· + (2 − 1) + (2 + 1) = 2 + (2 + 1) = ( + 1) 2
which shows that +1 is true.
• Therefore, by mathematical induction, 1 + 3 + 5 + ··· + (2 − 1) = 2 for every positive integer .
21. 0 () = 2 and +1 () = 0 ( ()) for = 012.
1 () = 0 ( 0 ()) = 0 2 =
2 2
= 4 , 2 () = 0 ( 1 ()) = 0 ( 4 ) = ( 4 ) 2 = 8 ,
3 () = 0 ( 2 ()) = 0 ( 8 ) = ( 8 ) 2 = 16 ,.Thus, a general formula is () = 2
+1 .
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
CHAPTER 1 PRINCIPLESOF PROBLEMSOLVING
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73
22. (a) 0 () = 1(2 − ) and +1 = 0 ◦ for = 012.
1 () = 0
1
2 −
=
1
2 −
1
2 −
=
2 −
2(2 − ) − 1
=
2 −
3 − 2 ,
2 () = 0
2 −
3 − 2
=
1
2 −
2 −
3 − 2
=
3 − 2
2(3 − 2) − (2 − )
=
3 − 2
4 − 3 ,
3 () = 0
3 − 2
4 − 3
=
1
2 −
3 − 2
4 − 3
=
4 − 3
2(4 − 3) − (3 − 2)
=
4 − 3
5 − 4
Thus, we conjecture that the general formula is () =
+ 1 −
+ 2 − ( + 1) .
To prove this, we use the Principle of Mathematical Induction. We have already verified that is true for = 1.
Assume that the formula is true for = ; that is, () =
+ 1 −
+ 2 − ( + 1) . Then
+1 () = ( 0 ◦ )() = 0 ( ()) = 0
+ 1 −
+ 2 − ( + 1)
=
1
2 −
+ 1 −
+ 2 − ( + 1)
=
+ 2 − ( + 1)
2[ + 2 − ( + 1)] − ( + 1 − )
=
+ 2 − ( + 1)
+ 3 − ( + 2)
This shows that the formula for is true for = + 1. Therefore, by mathematical induction, the formula is true for all
positive integers .
(b) From the graph, we can make several observations:
• The values at each fixed = keep increasing as increases.
• The vertical asymptote gets closer to = 1 as increases.
• The horizontal asymptote gets closer to = 1
as increases.
• The intercept for +1 is the value of the
vertical asymptote for .
• The intercept for is the value of the
horizontal asymptote for +1 .
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.