Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
11.
d = s −s
n
n−1
n n−1
( ) ( )
= ln 3 −ln 3
nln ( 3) ( n 1) ln ( 3)
( ln 3 )( n ( n 1 )) ( ln 3)( n n 1)
= − −
= − − = − +
= ln 3
The difference between consecutive terms is
constant, therefore the sequence is arithmetic.
s
1
= ln 3 = ln 3 , s
2
= ln 3 = 2ln 3 ,
s
( ) ( ) ( ) ( )
3 4
( ) ( ) s ( ) ( )
1 2
= ln 3 = 3ln 3 , = ln 3 = 4ln 3
3 4
ln n ln( n−1)
12. d s s e e n ( n )
= n − n−1 = − = − − 1 = 1
The difference between consecutive terms is
constant, therefore the sequence is arithmetic.
ln1 ln 2 ln 3
1 2 3
ln 4
4 = e = 4
s = e = 1, s = e = 2, s = e = 3,
s
13. an
= a1 + ( n−1)
d
= 2 + ( n −1)3
= 2+ 3n
−3
= 3n
−1
a 51 = 351 ⋅ − 1=
152
14. an
= a1 + ( n−1)
d
=− 2 + ( n −1)4
=− 2+ 4n
−4
= 4n
−6
a 51 = 451 ⋅ − 6=
198
15. an
= a1 + ( n−1)
d
= 5 + ( n −1)( −3)
= 5− 3n
+ 3
= 8−3n
a 51 = 8−3⋅ 51=−
145
16. an
= a1 + ( n−1)
d
= 6 + ( n −1)( −2)
= 6− 2n
+ 2
= 8−2n
a 51 = 8 −2⋅ 51 =− 94
17.
an
= a1 + ( n−1)
d
1
= 0 + ( n −1) 2
1 1
= n −
2 2
1
= ( n −1)
2
1
a 51 = ( 51 − 1 ) = 25
2
18. an
= a1 + ( n−1)
d
⎛ 1 ⎞
= 1 + ( n −1)
⎜ − ⎟
⎝ 3 ⎠
1 1
= 1− n + 3 3
4 1
= − n
3 3
4 1 4 51 47
a 51 = − ⋅ 51 = − =−
3 3 3 3 3
19. an
= a1 + ( n−1)
d
= 2 + ( n −1) 2
= 2+ 2n− 2 = 2n
a 51 = 51 2
20. a a n d n ( n )
n = 1 + ( − 1) = 0 + ( −1) π= −1
π
a 51 = 51π−π= 50π
21. a1 = 2, d = 2, an
= a1+ ( n−
1) d
a 100 = 2 + (100 − 1)2 = 2 + 99(2) = 2 + 198 = 200
22. a1 = − 1, d = 2, an
= a1+ ( n−
1) d
a 80 = − 1 + (80 − 1)2 = − 1+ 79(2) =− 1+ 158 = 157
23. a1 = 1, d =−2 − 1 =− 3, an
= a1+ ( n−
1) d
a 90 = 1 + (90−1)( − 3) = 1+ 89( −3)
= 1− 267= −266
24. a1 = 5, d = 0 − 5 =− 5, an
= a1+ ( n−
1) d
a 80 = 5 + (80−1)( − 5) = 5+ 79( −5)
= 5 − 395 = −390
1248
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