Precalculus
Unit 9 – Sequences, Series, & Binomial Theorem
Name
Date
Friday May 15, 2015 + one more day =
Homework: Review #3 2015
1.
We have used several formulas for these arithmetic and geometric sequences.
Please write them in the appropriate places without using your notes.
Recursive form for an arithmetic sequence
Explicit form for nth term of an arithmetic sequence
or
Sum of arithmetic sequence
Recursive form for a geometric sequence
Explicit form for nth term of a geometric sequence
Sum of a finite geometric sequence
Sum of an infinite geometric sequence if r < 1
2.
For each sequence complete the following tasks.
• Determine if the sequence is arithmetic, geometric, or neither. Identify the common difference or
common ratio as appropriate. Use proper notation and show supportive work.
• Write the nth term of each sequence using an explicit formula.
• Determine if the sum of the series would converge or diverge.
a. -16, -11, -6, -1, 4, …
an =
b. 4, 16, 64, 256, 1024, ...
an =
c. -2, 1, 6, 13, 22, …
an =
d. -0.75, 3, -12, 48, -192, …
an =
2 2
e. 18, -6, 2, − , , …
3 9
an =
#3– 7: Find the first four terms of the sequence. Use proper notation and show supportive work.
3.
an =
2n + 1
n+3
4.
an =
6.
ak =
1
a k −1 + 1, a1 = − 6
2
7.
ak +1 = −3 ( ak ) , a1 = 2
n!
2n
5. an = (−1) n (2n − 3)
2
Write an equation for the nth term using an explicit rule. You MUST show appropriate work and use good
notation to receive full credit. (Reminder: Write formulas in their generic form first.)
8.
a13 = 242 and a32 = 622 (arithmetic)
9.
a12 = 67 and a31 = 200 (arithmetic)
10.
a3 = −16 and a6 = 128
11.
a4 =
(geometric)
1
1
and a6 =
2
8
(geometric)
12. The sixth and eleventh terms of an arithmetic sequence are 20 and 45 respectively. Find the recursive rule.
13. The fifth and twelfth terms of an arithmetic sequence are 51 and 23 respectively. Find the recursive rule.
14. The fifth and eleventh terms of a geometric sequence are –486 and –354294 respectively. Find the recursive rule.
15. The third and sixth terms of a geometric sequence are –16 and 128 respectively. Find the recursive rule.
Find each sum. You MUST show appropriate work and use good notation to receive full credit. (Reminder:
Write formulas in their generic form first.) Use fractional form if necessary and appropriate.
30
∑( 5n − 5)
16.
50
17.
n=1
10
18.
∑ −3( −2)
n−1
8
19.
n=1
∞
20.
⎛1⎞
−2 ⎜ ⎟
∑
⎝5⎠
k =1
∑ ( 6n + 4)
n=11
∑5
k −1
k =1
k −1
∞
21.
⎛ 1⎞
9⎜ − ⎟
∑
m =1 ⎝ 3 ⎠
m −1
#22 – 25: Write the series in sigma notation.
22. 1 + 4 + 16 + 64 + . . . + 65536
23.
1
4
1
+ 12
+
24. −5 + 4 + 13 + 22 + . . . + 85
25.
2
5
+ 64 + 78 + 168 + . . . + 256
12
1
36
1
+ 108
+. . .+
1
2916
Find each sum. You MUST show appropriate work and use good notation to receive full credit. (Reminder:
Write formulas in their generic form first.) Use fractional form if necessary and appropriate.
26.
S8 ;
28.
S9 3, 6, 12, 24,
30.
4+6+9+
4, 8, 16, 32, …
27 81
+ + ...
2 2
27.
S80 ,
11, 17, 23, 29…
29.
6+3+
31.
14 + 8 + 2 + ( −4) + ...
3 3 3
+ + + ...
2 4 8
You MUST show appropriate work and use good notation to receive full credit. (Reminder: Write formulas in their generic form first.)
32. Blazer auditorium has 18 seats in the first row. Each successive row has 3 additional seats. The last row has 90
seats.
a.
Determine how many rows are in the auditorium.
b. Find the total number of seats in the auditorium.
33. Another auditorium has 32 rows with 18 seats in the first row and 4 more seats in each of the successive rows.
How many seats are there in this auditorium?
34. A ball is dropped from a height of 8 feet. The ball rebounds to 80% of its previous height with each bounce. How
high (to the nearest inch) does the ball rebound on the fifth bounce?
35. A ball rebounds each time to a height equal to one half the height of the previous bounce. If the ball is dropped
from a height of 16 m find the total distance it has travelled when it hits ground for the 10th time. Round your
final answer to the nearest tenth of a meter.
36. A patient is given a 20 mg injection of a therapeutic drug each day. The patient’s body metabolizes 50% of the
drug present, so that after 1 day only one-half of the original amount remains. The patient is given a 20mg
injection of the drug every day at the same time. What quantity of the drug remains in the patient’s body after the
10th injection?