Chapter 12 Sequences; Induction; the Binomial Theorem
Chapter 12 Sequences; Induction; the Binomial Theorem
Chapter 12 Sequences; Induction; the Binomial Theorem
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<strong>Chapter</strong> <strong>12</strong>: <strong>Sequences</strong>; <strong>Induction</strong>; <strong>the</strong> <strong>Binomial</strong> <strong>Theorem</strong><br />
34. <strong>12</strong> 1 18 1<br />
a = a + 11d = 4 a = a + 17d<br />
= 28<br />
Solve <strong>the</strong> system of equations by subtracting <strong>the</strong><br />
first equation from <strong>the</strong> second:<br />
6d<br />
= 24⇒ d = 4<br />
a = 4− 11(4) = 4− 44= −40<br />
1<br />
Recursive formula: a1 =− 40 an<br />
= an<br />
− 1+<br />
4<br />
a = a + n−<br />
d<br />
nth term: n 1 ( 1)<br />
=− 40 + ( n −1)( 4)<br />
=− 40 + 4n<br />
−4<br />
= 4n<br />
−44<br />
n n n<br />
1 n 1 2 1 2<br />
2 2 2<br />
2<br />
35. S = ( a + a ) = ( + ( n− )) = ( n) = n<br />
n<br />
n n<br />
1 n 2 2 1<br />
2 2<br />
2<br />
36. S = ( a + a ) = ( + n) = n+ n = n( n+<br />
)<br />
n<br />
n n n<br />
1 n 7 2 5 9 5<br />
2 2 2<br />
37. S = ( a + a ) = ( + ( + n)<br />
) = ( + n)<br />
n<br />
n n<br />
2<br />
1 n<br />
2<br />
n<br />
2<br />
= ( 4n− 6)<br />
= 2n −3n<br />
2<br />
= n −<br />
38. S = ( a + a ) = ( − 1+ ( 4n−5)<br />
)<br />
n<br />
( 2n<br />
3)<br />
39. a1 = 2, d = 4 − 2 = 2, an<br />
= a1+ ( n−<br />
1) d<br />
70 = 2 + ( n −1)2<br />
70 = 2 + 2n<br />
−2<br />
70 = 2n<br />
n = 35<br />
n 35 35<br />
Sn<br />
= ( a1<br />
+ an) = ( 2+ 70) = ( 72)<br />
= <strong>12</strong>60<br />
2 2 2<br />
40. a1 = 1, d = 3 − 1 = 2, an<br />
= a1+ ( n−<br />
1) d<br />
59 = 1 + ( n −1)2<br />
59 = 1+ 2n<br />
−2<br />
60 = 2n<br />
n = 30<br />
n 30<br />
Sn<br />
= ( a1<br />
+ an) = ( 1 + 59 ) = 15 ( 60 ) = 900<br />
2 2<br />
41. a1 = 5, d = 9 − 5 = 4, an<br />
= a1+ ( n−<br />
1) d<br />
49 = 5 + ( n −1)<br />
4<br />
49 = 5 + 4n<br />
−4<br />
48 = 4n<br />
n = <strong>12</strong><br />
n <strong>12</strong><br />
Sn<br />
= ( a1<br />
+ an) = ( 5 + 49 ) = 6 ( 54 ) = 324<br />
2 2<br />
42. a1 = 2, d = 5 − 2 = 3, an<br />
= a1+ ( n−<br />
1) d<br />
41 = 2 + ( n −1)<br />
3<br />
41 = 2 + 3n<br />
−3<br />
42 = 3n<br />
n = 14<br />
n 14<br />
Sn<br />
= ( a1<br />
+ an) = ( 2 + 41 ) = 7 ( 43 ) = 301<br />
2 2<br />
43. a 1 = 73 , 78 73 5<br />
d = − = ,<br />
n<br />
=<br />
1<br />
+ ( − 1)<br />
( n )( )<br />
( n )<br />
558 = 73 + −1 5<br />
485 = 5 −1<br />
97 = n −1<br />
98 = n<br />
n 98<br />
Sn<br />
= a1<br />
+ an<br />
= 73 + 558<br />
2 2<br />
= 49 631 = 30,919<br />
( ) ( )<br />
( )<br />
44. a 1 = 7 , 1 7 6<br />
a a n d<br />
d = − = − ,<br />
n = 1 + ( − 1)<br />
( n )( )<br />
( n )<br />
− 299 = 7 + −1 −6<br />
− 306 = −6 −1<br />
51 = n −1<br />
52 = n<br />
n 52<br />
Sn<br />
= a1<br />
+ an<br />
= 7 + − 299<br />
2 2<br />
= 26 − 292 = −7592<br />
a a n d<br />
( )<br />
( ) ( )<br />
( )<br />
45. a 1 = 4 , 4.5 4 0.5<br />
d = − = ,<br />
n<br />
=<br />
1<br />
+ ( − 1)<br />
( n )( )<br />
( n )<br />
100 = 4 + −1 0.5<br />
a a n d<br />
96 = 0.5 −1<br />
192 = n −1<br />
193 = n<br />
n 193 193<br />
Sn<br />
= a1<br />
+ an<br />
= 4+ 100 = 104<br />
2 2 2<br />
= 10,036<br />
( ) ( ) ( )<br />
<strong>12</strong>50<br />
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