Sequences & Series Problems
Name_________________________
The first 15 problems are simpler (and mostly all arithmetic), so PLEASE don't assume you know
everything if you get them all right.
1.
Find the sum of the 1st 500 terms of the series below:
6 + 18 + 30 + 42 + 54 + ...
2.
Write the explicit formula for the sequence below:
22 + 19 + 16 + 13 + 10 + ...
3.
What is the common difference in an arithmetic sequence if the value of the 36th term
is -78, and the value of the 100th term is 42?
4.
Consider a sequence with the recursive formula below:
a1  900; an  an1  35 .
What is the value of the 8th term of the sequence?
41
5.
Evaluate:
 9n  147
n 3
6.
Suppose a sequence has an explicit formula of an  4.5n  1 .
What is the value of the 102nd term of the sequence?
7.
What is the position (1st, 2nd, 3rd, etc.) of the term with a value of -11 in the sequence
with an explicit formula of an  15n  484 ?
8.
What is the value of the first term in an arithmetic sequence if the following is known:
a21  50 and a47  106 ?
9.
An arithmetic series has an explicit formula of an  9n  10 . What is the sum of all of the
terms from the 113th term thru the 125th term?
10.
Consider the explicit formula for all arithmetic sequences.
Which one of the following statements is FALSE?
A)the value of n can be negative
B)the value of d can be negative
C)the value of a1 can be negative
D)the value of d does not have to be a whole number (example: 0.3)
11.
Which of the following expressions in summation notation will NOT form a series
which is arithmetic?
5
A)
 10n
5
C)
B)
 400n  1
n 1
12.
3
6
n 1
n 1
5
n
5
D)
n
n 1
Consider this series: 5 + 10 + 15 + 20 + ...
Find the sum of all of the terms from the 22nd term thru the 43rd term.
13.
Find the sum of the entire finite series given below:
(-44) + (-37) + (-30) + ... + 404 + 411 + 418
14.
Write the explicit formula for an arithmetic sequence in which the value of the 42nd
term is -71, and the value of the 65th term is -117.
15.
The sum of the first 24 terms of an arithmetic series is 720. If the value of the 24 th term
is -79, what is the value of the 1st term?
16.
Consider this sequence: 14, -4, -22, -40, -58, ...
Write its explicit formula.
17.
Find the common ratio (r) for the geometric sequence below.
16, 60, 225, 843.75, 3164.0625, ...
18.
Suppose there is an arithmetic sequence in which the value of the 8th term is 10, and
the value of the 22nd term is 52. What is the value of the 2nd term of the sequence?
19.
Evaluate:
29
 5n  27
n4
20.
Find the sum of the 1st 63 terms of the sequence below:
105 + 101 + 97 + 93 + 89 + ...
21.
A geometric sequence has the following properties:
a3  12 and a6  96 .
Write the explicit formula for the geometric sequence.
22.
There exists an arithmetic sequence in which the value of the first term is -121, and the
common difference is 26. What is the position (1st, 2nd, 3rd, etc.) of the term in
the sequence with a value of 945?
23.
Consider the following expression:
3
 2n  1
n 1
Which of the following is the series represented by the expression above?
24.
A) (-1) + 1 + 3
C) (-3) + (-1) + 1
B) (1) + (2) + (3)
D) (-1) + (-3) + (-5)
Consider a sequence with an explicit formula of an  5  (3)n1 .
What is the position (1st, 2nd, 3rd, etc.) of the term with a value of 7,971,615?
25.
The sequence below is a geometric sequence. Find the value of the term located at the
blank with the arrow.
-2, _____, _____, _____, _____, 33,614
↑
26.
Consider the following series: 0 + 10 + 20 + 30 + 40 + ...
What is the sum of all of the terms from the 49th term thru the 77th term?
27.
Suppose a sequence has an explicit formula of an  16n  8 .
What is the value of the 200th term in the sequence?
28.
Consider a sequence with the recursive formula below:
a1  8; an  an1  41 .
What is the value of the 13th term of the sequence?
29.
Find the sum of the entire finite series given below:
306 + 303 + 300 ... + (-90) + (-93) + (-96)
30.
A geometric sequence exists in which the value of the first term is 7, and the value of
the 3rd term is 28.
I. The common ratio could be 10.5.
II. The common ratio could be 2.
III. The common ratio could be -2.
Which of the statements above is true?
A)I only
B)II only
C)III only
31.
D)Both I and II
E)Both II and III
Consider this arithmetic series: (-36) + (-32) + (-28) + (-24) + ...
Suppose the sum of the 1st n terms of this series is equal to 7632.
What is the value of n?
32.
Consider the following sequence: -1.2, 1.92, -3.072, 4.9152, ...
What is the common ratio (r) of the sequence?
33.
Suppose an arithmetic sequence has an explicit formula of an  4n  17 . What is the
sum of all of the terms from the 109th term thru the 115th term?
34.
There exists a geometric sequence in which the value of the 5th term is 80 and the value
of the 8th term is 5,120. Write the explicit formula for the sequence.
35.
The sum of the first n terms of a geometric series is -6,830. If the geometric series is
constructed so that the value of the first term is -10, and the common ratio is -2, what
is the value of n?
36.
Find the sum (if it exists) of the infinite series below:
20 + 15 + 11.25 + 8.4375 + ...
37.
Consider a sequence with the recursive formula below:
a1  7; an  an1  3 .
What is the value of the 8th term of the sequence?
8
38.
Evaluate:
 3(5)
k 1
k 1
39.
Find the sum of the entire finite sequence below:
11 + 25 + 39 + ... + 655 + 669 + 683.
40.
Which of the following infinite series will NOT have a sum?
A) 0.5 + 0.6 + 0.72 + 0.864 + ...
B) (-10) + (-5) + (-2.5) + (-1.25) + ...
C) 100,000 + 10,000 + 1,000 + 100 + ...
1
1
D) 4 + (-1) + + ( ) + ...
4
16
41.
Write the explicit formula for the sequence below:
-4, -14, -24, -34, -44, ...
42.
Consider a sequence with an explicit formula of an  64  (1.5)n1 .
What is the position (1st, 2nd, 3rd, etc.) of the term with a
value of 2460.375?
43.
Which of the following expressions in summation notation will NOT form a series
which is arithmetic?
10
10
A)
 n2  3n
C)
n 1
n 1
10
10
B)
 7n
D)
 4n  1000
n 2
n 1
44.
 n  5
An arithmetic sequence has the following properties:
a10  74 and a29  17 .
Find the value of a35 .
45.
An arithmetic sequence has an explicit formula of an  25n  1 .
What is the value of the 500th term of the sequence?
n
46.
Find the value of n in the expression to the right:
 6k  10  3230 .
k 1
47.
In an arithmetic sequence, suppose that the value of the 17th term is 106, and the value
of the 40th term is -147. What is the value of the common difference (d)?
48.
Evaluate:

 46(0.8)
k 1
k 1
49.
50.
The explicit formulas for four different sequences are given. Which formula
corresponds to the sequence in which a9  768 ?
A) an  40n  106
C) an  4  (3)n1
B) an  9n  768
D) an  3  (2)n 1
Find the sum of the 1st 10 terms of the sequence below:
1.5 + 6 + 24 + 96 + ...
51.
Consider a sequence with this explicit formula: an  2048  (0.5)n1 .
What is the position (1st, 2nd, 3rd, etc.) of the term in the
sequence with a value of -0.25?
52.
A sequence has the following characteristics:
a1  57 ; an  an1  6 .
What is the value of the 98th term of the sequence?
53.
Consider this arithmetic series: (-18) + (-15) + (-12) + (-9) + ...
Suppose the sum of the 1st n terms of the series is equal
to 7,821. What is the value of n?
54.
If the sequence below is GEOMETRIC, find the value of the term indicated by the arrow.
Assume r is positive.
______, 42, _____, _____, _____, 100,842
↑
54
55.
Evaluate:
 5n  4 .
n 25
56.
Write the explicit formula for the sequence below:
5, -3, -11, -19, -27, ...
57.
A finite geometric series has a first term with a value of 5 and a common ratio of 2. If
the sum of all of the terms is equal to 655,355, then how many terms are in the series?
58.
Which sequence below is NOT arithmetic?
A)1, 2, 4, 7, 11, 16, ...
B)2, 4, 6, 8, 10, ...
C)6, -13, -32, -51, -70, ...
D)-6, -3, 0, 3, 6, ...
23
59.
Evaluate:
 2.25(2)
m 1
m5
60.
Consider a geometric sequence in which the value of the 3rd term is 10, and the value of
the 9th term is 156250. Write the common ratio and the explicit formula for this
sequence. Assume the value of r is positive.
61.
In an arithmetic series, suppose the sum of the 1st 60 terms is 11910. If the value of the
60th term is 405, what is the value of the 1st term?
62.
How many terms are in the sequence below?
34, 16, -2, -20, ..., -1298, -1316, -1334
63.
An infinite series is given in the space below.
What is the sum of all of the terms in the series?
5 + 15 + 45 + 135 + ...
64.
A sequence has the following characteristics:
a1  3 ; an  an1  4 .
What is the value of the 10th term of the sequence?
n
65.
Find the value of n in the expression:
 12k  45  77571 .
k 1
66.
Consider a sequence with this explicit formula: an  2n  65 .
What is the position (1st, 2nd, 3rd, etc.) of the term in the
sequence with a value of -999?
67.
Consider the following series: 512 + 768 + 1152 + 1728 + ...
Suppose the sum of the 1st n terms of this series is 87,549.5.
What is the value of n?
68.
Consider the following statements about an arithmetic sequence:
I. The value of n can be negative.
II. The value of n can be a decimal number (ex. 4.5, 1.77).
III. The value of an can be negative.
Which of the statements are true?
A)I only
B)II only
C)III only
D)Both I and II
E)Both II and III
69.
Here is a type of sequence you have never seen: 4, 7, 12, 19, 28, ...
What is its explicit formula?
70.
A) an  n2  3
C) an  2n  2
B) an  3n  1
D) an  n3  3n
What is the common difference of the sequence below?
-32, -23, -14, -5, ...
1. 1,500,000
2. an  3n  25
3. 1.875
4. 655
5. -1989
6. 460
7. 33
8. 170
9. 13,793
10. A
11. C
12. 3575
13. 12,529
14. an  2n  13
15. 139
16. an  18n  32
17.
18.
19.
20.
3.75
-8
2847
-1197
21. an  3(2)n1
22.
23.
24.
25.
26.
27.
28.
29.
30.
42
D
14
686
17,980
-3192
500
14,175
E
31. 72
32. -1.6
33. 3255
34. an  .3125(4)n1
35.
36.
37.
38.
39.
40.
41.
11
80
15,309
-292,968
17,003
A
an  10n  6
42. 10
43. A
44. -1
45. 12,501
46. 34
47. -11
48. 230
49. D
50. 524287.5
51. 14
52. 525
53. 79
54. 6
55. 6045
56. an  8n  13
57. 17
58. A
59. 18,874,332
60. r=5; an  .4(5)n1
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
-8
77
no sum
-786,432
117
467
11
C
A
9