Section 5.1 Homework Solutions
Write the first four terms of the sequences defined below:
1. a k 
k
, for all integers k  1.
10  k
3. ci 
(1) i
, for all integers i  0 .
3i
n
5. en     2 , for all integers n  0 .
2
Compute the first fifteen terms of the following sequence, and give a description in words of its general behavior:
8. g n  log 2 n , for all integers n  1.
Find explicit formulas for the following sequences:
10. -1, 1, -1, 1, -1, 1, …
11. 0, 1, -2, 3, -4, 5, …
13. 1 
1 1 1 1 1 1 1 1 1
,  ,  ,  ,  ,
2 2 3 3 4 4 5 5 6
16. 3, 6, 12, 24, 48, 96, …
Compute the summations and products:
5
19.
 (k  1) 
k 1
4
20.
k
k 2
2

1
23.
 i(i  1) 
i 1
Evaluate the summations and products for the indicated values of the variable.
33.
1
1
1
1
1
 2  2  ...  2 ; n  1 is just 2  1
2
1
2
3
n
1
 1  2  3 
 k 


...  
; k  3
 1  1  2  1  3  1 
 k  1
35. 
 1  2  3  1 2 3 1



.    
 1  1  2  1  3  1  2 3 4 4
Rewrite by separating off the final term.
37.
39.
k 1
k
i 1
i 1
 i(i!)   i(i!)  (k  1)  (k  1)!
n 1
n
m 1
m 1
 m(m  1)  m(m  1)  (n  1)  (n  2)
Write each of the following using summation or product notation:
43. 12  2 2  32  4 2  5 2  6 2  7 2 
46.
2
3
4
5
6





3 4 45 5 6 6 7 7 8
51. n  (n  1)  (n  2)    2  1 
or
n  (n  1)  (n  2)    2  1 
Transform using the change of variable i  k  1 :
5
53.
 k (k  1) 
k 0
Transform using the change of variable j  i  1 :
n 1
57.
i
 (n  i )
i 1
2

Write as a single summation:
n
n
k 1
k 1
 (2k  3)   (4  5k ) 
59. 3
Compute:
63.
6!

8!
66.
(n  1)!

(n  1)!
 5
5!
5!
54
3
3!
3!
3!


 10
71.   
2
 3  3!(5  3)! 3!2!

 1
73.   
 0  0!(3  0)! 1  3! 3!
 5
5!
5!
5!
74.   


1
 5  5!(5  5)! 5!0! 5!1
 n  1
(n  1)!
(n  1)!
(n  1)( n)
 


2
 n  1 (n  1)!(( n  1)  (n  1))! (n  1)!2!
76. 