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Section 5.1 Homework Solutions Write the first four terms of the sequences defined below:
1. k
kak +=
10, for all integers 1≥k .
3. i
i
ic3
)1(−= , for all integers 0≥i .
5. 22
⋅
=nen , for all integers 0≥n .
Compute the first fifteen terms of the following sequence, and give a description in words of its general behavior: 8. ngn 2log= , for all integers 1≥n . Find explicit formulas for the following sequences: 10. -1, 1, -1, 1, -1, 1, … 11. 0, 1, -2, 3, -4, 5, …
13. ,61
51,
51
41,
41
31,
31
21,
211 −−−−−
16. 3, 6, 12, 24, 48, 96, … Compute the summations and products:
19. =+∑=
5
1)1(
kk
20. =∏=
4
2
2
k
k
23. =+∑=
1
1)1(
iii
Evaluate the summations and products for the indicated values of the variable.
33. 1;1...31
21
11
2222 =++++ nn
is just 1112 =
35. 3;1
...13
312
211
1=
++
+
+
+
kk
k
41
43
32
21.
133
122
111
=⋅⋅=
+
+
+
Rewrite by separating off the final term.
37. )!1()1()!()!(1
1
1+⋅++= ∑∑
=
+
=
kkiiiik
i
k
i
39. ∑∑=
+
=
+⋅+++=+n
m
n
mnnmmmm
1
1
1)2()1()1()1(
Write each of the following using summation or product notation: 43. =+−+−+− 2222222 7654321
46. =⋅
+⋅
−⋅
+⋅
−⋅ 87
676
565
454
343
2
51. =+++−+−+ 12)2()1( nnn or
=+++−+−+ 12)2()1( nnn
Transform using the change of variable 1+= ki :
53. ∑=
=−5
0)1(
kkk
Transform using the change of variable 1−= ij :
57. =−∑
−
=
1
12)(
n
i ini
Write as a single summation:
59. =−+− ∑∑==
n
k
n
kkk
11)54()32(3
Compute:
63. =!8!6
66. =+−
)!1()!1(
nn
71. 10245
!2!3!5
)!35(!3!5
35
=⋅
=⋅
=−
=
73. 1!3!3
!31!3
)!03(!0!3
03
==⋅
=−
=
74. 11!5!5
!0!5!5
)!55(!5!5
55
=⋅
=⋅
=−
=
76. 2
))(1(!2)!1(
)!1())!1()1(()!1(
)!1(11 nn
nn
nnnn
nn +
=⋅−
+=
−−+−+
=
−+
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