Pre-Calculus 11
Unit 1: Sequences & Series – Review
Multiple Choice Section
1. Which of the following is a geometric sequence?
A. 5 , 3 , 2 , 1
B. 6 , 4 , 2 , 0
C. 7 , 4 , 2 , 1
D. 8 , 4 , 2 , 1
2. Given the arithmetic series with first term 3 and common difference 4, determine t 7 .
A. 4099
B. 27
C. 24
D. 31
2
4
3. Determine the common ratio of the geometric sequence: 3, 5 , 75 , …
2
A. 15
3
B. 10
2
6
C. 5
D. 5
4. Determine the 14𝑡ℎ term of the geometric series: 6 + 12 + 24 + …
A. 12 288
B. 24 576
C. 49 152
5. Determine the common ration of the geometric sequence: 1,
A. -3
B.
−1
3
1
−1 1
3
D. 98 304
−1
, 9 , 27
C. 3
D. 3
6. Given the arithmetic series 5  2 1  ... , determine the sum of the first 10 terms.
A. –85
B. –185
D. –220
C. 185
7. Find the sum of the first 20 terms of the geometric sequence 125, 100, 80, 64, …
A. 2 250
B. 274.89
C. 617.79
D. 42 868.09
9
8. Find the sum of the infinite geometric series: 8 − 6 + 2 −
A. –32
32
B. – 7
C.
32
27
8
+⋯
7
9. Find the sum of the infinite geometric series: 175 − 70 + 28 − ⋯
A. 125
B. 133
C. 291.67
D. 32
D. no finite sum
10. While training for a race, a runner increases her distance by 10% each day. If she runs
2km on the first day, what will be her total distance for 26 days of training? (Accurate to
2 decimal places.)
A. 21.67km
B. 23.84km
C. 196.69km
D. 218.36km
11. If an arithmetic series has S9  57.6 and t1  2 , then determine the common difference.
A. –0.1
B. 1.1
C. 0.97
D. 0.55
12. Determine the number of terms in the following geometric sequence:
𝑎2
𝑏14
, 𝑎, 𝑏, … , 14
𝑏
𝑎
A. 14
B. 15
C. 16
D. 17
13. In a geometric sequence, 𝑡1 = 16 and 𝑡5 = 81. Find 𝑡7 .
A. 107
B. 121.5
C. 182.25
D. 237.39
14. Which term in the arithmetic sequence 3, 1.1, 0.8,... is 19.8?
A. 11th
B. 12th
C. 13th
D. 14th
15. Determine the common difference for the arithmetic sequence: 1.6, 2.8, ...
A. 1.6
B. 2.8
C. 1.2
D. –4.4
16. In a geometric sequence, 𝑡3 = 256 and 𝑡8 = 781.25. Find the common ratio r.
A. 1.20
B. 1.25
C. 1.32
D. 3.50
17. In a geometric sequence, the 9𝑡ℎ term is –5 and the 12𝑡ℎ term is 40. Determine the
common ratio of this sequence.
1
1
A. –8
B. –2
C. – 2
D. – 8
18. Which term of the geometric sequence 5 , 15 , 45 , … is 885 735?
A. 10𝑡ℎ
B. 11𝑡ℎ
C. 12𝑡ℎ
D. 13𝑡ℎ
19. Determine the common difference for the arithmetic sequence: 2 x, 6 x, 10 x, ...
A. 4
B. 4x
C. x
D. 3x
20. Given that an arithmetic series has t2  12 and S2  22 , determine the common
difference.
A. 2
B. 10
C. 12
D. 14
21. Given the geometric sequence 𝑏 2 , 𝑏 2.25 , 𝑏 2.5 , 𝑏 2.75, 𝑏 3 , … , determine an expression for
the 𝑛𝑡ℎ
A. 𝑏 0.25𝑛 + 1.75
B. 𝑏 2.25−0.25𝑛
C. 𝑏 0.25𝑛 + 2
D. 𝑏 𝑛−0.25
22. For a geometric sequence, 𝑡7 = 5x + 2 and 𝑡10 = x – 23. If the common ratio r is 2,
find 𝑡10 .
A. –26
B. –24
C. –12
D. –3
23. The second term of a geometric series is –16 and the seventh term is 512. Determine the
first term.
A. –2
B. 2
C. –8
D. 8
24. If x , 4 , 8x are three consecutive terms in a geometric sequence, determine the values of x.
A. ±1
B. ±√2
C. ±2
D. ±2√2
25. If the sum of the first 5 terms of a geometric series is -328 and the common ratio is −4,
determine the first term.
A. –3.86
B. –1.6
C. 0.96
D. 6.43
26. An aquarium originally containing 30 litres of water loses 6% of its water to evaporation
every day. Determine a geometric sequence which shows the number of litres of water in
the aquarium on consecutive days.
A. 30, 30(0.94), 30(0.94)2 , 30(0.94)3 , …
B. 30, 30(0.06), 30(0.06)2 , 30(0.06)3 , …
30
30
30
30
30
30
C. 30, 1.06 , (1.06)2 , (1.06)3
D. 30, 0.94 , (0.94)2 , (0.94)3
27. Given a geometric series with a first term of 14 and a common ratio of 1.8, determine the
sum of the first 10 terms.
A. –4 986.45
B. 2 777.03
C. 2 453.79
D. 6 230.82
28. A ball is dropped from a height of 5m. After each bounce, it rises to 60% of its previous
height. What is the total vertical distance the ball travels before it comes to rest?
A. 12.5m
B. 15m
C. 20m
D. 25m
1
29. For what values of 𝑥 , 𝑥 ≠ 2, will the following infinite geometric series have a finite
sum?
1 + (2𝑥 − 1) + (2𝑥 − 1)2 + (2𝑥 − 1)3 + ⋯
A. −1 < 𝑥 < 0
B. −1 < 𝑥 < 1
1
3
C. − 2 < 𝑥 < 2
D. 0 < 𝑥 < 1
2
30. The 𝑛𝑡ℎ term of an infinite series is given by 𝑡𝑛 = 5(3)𝑛−1 . Find the sum of the series.
A. 3
B. 10
C. 15
D. no finite sum
1
31. An infinite geometric series has a finite sum and a common ratio 𝑟 = 𝑥−1. Which of the
following could not be a value for 𝑥?
A. –0.6
B. –0.5
C. 2
D. 3
32. In an arithmetic sequence, the 5th term is 12 and the 8th term is 18.9. Determine the
common difference of this sequence.
A. 3.45
B. 6.9
C. 2.3
D. 1.16
33. The sum of an infinite geometric series is 15. If the 1𝑠𝑡 term is 6, find the common ratio r.
3
2
A. − 5
2
B. − 5
𝑘+2
34. Which of the following is equivalent to ∑20
?
𝑘=1 3
A.
27(1−320 )
B.
−2
3
C. 5
27(1−319 )
C.
−2
63
35. Determine the number of terms in the series:
D. 5
3(1−320 )
 3 2
D.
−2
3(1−322 )
−2
k 1
k  21
A. 21
B. 42
C. 43
D.63
Open ended section
36. Find the sum of the first 10 terms of the series:
3
9
27
a) 1 − + − + ⋯
4
16
64
b) √2 + 2 + √8 + ⋯
37. Given the arithmetic series 2 x, 3x  3, 4 x  6,...
a) Determine the 7th term.
b) Which term in the series is 16x  42 ?
S
c) Determine an expression for the partial sum 8 of this series.
d) If the 5th term of the series is –18, then determine the common difference.
38. A ball dropped from 10 m rebounds ¾ of the distance from where it fell.
a) Find the total vertical distance that the ball travels before coming to rest.
b) Find the total vertical distance that the ball travels after 10 bounces.
c) What is the total distance travelled after 10th bounce?
d) A weather balloon rises 100m the first minute, and each minute later it rises 4% less
than the previous minute. What is the maximum height reached by the balloon?
39. Determine the partial sum of the arithmetic series with t1  5.8 and last term t12  17.7
40. A weather balloon rises 100 m the first minute, and each minute later it rises 4% less than
the previous minute. What is the maximum height reached by the balloon?
41. Determine algebraically (using a series) what the following repeating decimals are in
fraction form.
a) 0.5
b) 0.23
c) 3.51
42. Make sure you know how to prove the equations for sequences and series.
Solutions
1. D
2. B
3. A
4. C
5. B
6. A
7. C
8. C
9. A
10. D
11. B
12. D
13. C
14. C
15. D
16. B
17. B
18. C
19. B
20. A
21. A
22. B
23. D
24. B
25. B
26. A
27. D
28. C
29. D
30. C
31. C
32. C
33. D
34. A
35. C
36. a) 0.539
b) 62 + 31√2
37. a) 8x 18
b) 15th
c) 44 x  84
d) –2
38. Ignore the initial 10 m in the infinite series, a = 15 r = ¾
a.) infinite sum = 60 m, total vertical distance = 70 m
b.) S10 = 56.62 m + 10 = 66.62 m
c.) 𝑆∞ − 𝑆10 = 3.38 𝑚
39. 71.4
40. 2 500 m
41. a)
5
9
b)
7
30
c)
116
33