5128_Ch08_434-471

Section 8.1 Sequences 441
Quick Review 8.1 (For help, go to Sections 1.2, 2.1, and 2.2.)
x
In Exercises 1 and 2, let f (x) . Find the values of f.
x 3
1. f (5) 58 2. f (2) 2
In Exercises 3 and 4, evaluate the expression a (n 1)d for the
given values of a, n, and d.
3. a 2, n 3, d 1.5 1
4. a 7, n 5, d 3 5
In Exercises 5 and 6, evaluate the expression ar n1 for the given
values of a, r, and n.
5. a 1.5, r 2, n 4 12 6. a 2, r 1.5, n 3 –4.5
In Exercises 7–10, find the value of the limit.
5x3
2x2
7. lim x→ 3 x4
0
8. lim
16x2
sin (3x)
3
x→0 x
9. lim
x→ x sin 1 x 1
10. lim 2x 3
x
2
x→ x 1
Does not exist, or
Section 8.1 Exercises
In Exercises 1–4, find the first six terms and the 50th term of the
sequence with specified nth term.
n
1. a n See page 443. 2. b n 3 1 See page 443.
n 1
n
3. c n 1 1 n n See page 443. 4. d n n 2 3n
2, 2, 0, 4, 10, 18; 2350
In Exercises 5–10, find the first four terms and the eighth term of the
recursively defined sequence.
5. a 1 3, a n a n1 2 for all n 2 3, 1, 1, 3; 11
6. b 1 2, b n b n1 1 for all n 2 2, 1, 0, 1; 5
7. c 1 2, c n 2c n1 for all n 2 2, 4, 8, 16; 256
8. d 1 10, d n 1.1d n1 for all n 2
10, 11, 12.1, 13.31; 19.487171 10(1.1) 7
9. u 1 1, u 2 1, u n u n1 u n2 for all n 3
1, 1, 2, 3; 21
10. v 1 3, v 2 2, v n v n1 v n2 for all n 3 3, 2, 1, 1; 2
In Exercises 11–14, the sequences are arithmetic. Find
(a) the common difference,
(b) the eighth term,
(c) a recursive rule for the nth term, and
(d) an explicit rule for the nth term.
11. 2, 1, 4, 7, … See page 443. 12. 15, 13, 11, 9, … See page 443.
13. 1, 32, 2, 52, … See page 443. 14. 3, 3.1, 3.2, 3.3, … See page 443.
21. The fourth and seventh terms of a geometric sequence are 3010
and 3,010,000, respectively. Find the first term, common ratio,
and an explicit rule for the nth term. a 1 3.01, r = 10,
a n 3.01(10) n1 , n 1
22. The second and seventh terms of a geometric sequence are 12
and 16, respectively. Find the first term, common ratio, and an
explicit rule for the nth term. a 1 14, r = –2,
a n (1) n1 (2) n3 , n 1
In Exercises 23–30, draw a graph of the sequence {a n }.
n
23. a n n
2
, n 1, 2, 3, …
1
24. a n n 2
, n 1, 2, 3, …
n 2
25. a n (1) n 2n 1
, n 1, 2, 3, …
n
26. a n 1 2 n n , n 1, 2, 3, …
27. u 1 2, u n 3u n1 for all n 2
28. u 1 2, u n u n1 3 for all n 2
29. u 1 3, u n 5 1 2 u n1 for all n 2
30. u 1 5, u n u n1 2 for all n 2
In Exercises 31–40, determine the convergence or divergence of the
sequence with given nth term. If the sequence converges, find its limit.
31. a n 3n 1
2n
converges, 3 32. a n converges, 2
n
n 3
In Exercises 15–18, the sequences are geometric. Find
(a) the common ratio,
33. a n 2 n2
n 1
n
5n2
34. a n
n 2
n
2
converges, 0
1
(b) the ninth term,
35. a n (1) n n converges, 25
1
diverges 36. a n (1) n n 1
(c) a recursive rule for the nth term, and
n 3
n
2
converges, 0
2
(d) an explicit rule for the nth term.
37. a n (1.1) n diverges 38. a n (0.9) n converges, 0
15. 8, 4, 2, 1, … See page 443. 16. 1, 1.5, 2.25, 3.375, … See page 443. 39. a n n sin
17. 3, 9, 27, 81, … 18. 5, 5, 5, 5, …
1 n converges, 1 40. a
n cos n 2 diverges
See
page 443.
See page 443.
In Exercises 41–44, use the Sandwich Theorem to show that the
19. The second and fifth terms of an arithmetic sequence are 2
sequence with given nth term converges and find its limit.
and 7, respectively. Find the first term and a recursive rule for
the nth term. –5, a n a n1 3 for all n 2
41. a n sin n
1 1
0 42. a n 0 (Note:
n
2 n
2 n 1 for n 1)
n
20. The fifth and ninth terms of an arithmetic sequence are 5 and
3, respectively. Find the first term and an explicit rule for the
1
43. a n 44. a n sin 2
n
0
nth term. 13, a n 2n 15, n 1
n !
2n
1
0, (Note: 1 for n 1)
n ! n