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 +

=⋅−

+=

−−+−+

=

−+