Mu Alpha Theta National Convention: Hawaii, 2005
Sequences and Series Topic Test – Theta Division
1.
Find the common difference of the arithmetic progression 2k, 4k + 1, 6k + 2, …
A) 2k + 1
2.
2
3
511
3
D) 630
E) NOTA
B)
1
6
C)
4 2
 
3 9
1
3
D)
1
2
E) NOTA
B) 992
C) 496
D) 528
E) NOTA
B) 146
C) -166
D) 150
B)
1
2
C) 341
D)
1 2 4
  
3 3 3
55
3
E) NOTA
?
E) NOTA
Evaluate the sum of the terms of the arithmetic sequence 40, 35, …, -95, -100.
A) 840
8.
C) 660
What is the sum of the first 10 terms of the geometric series
A)
7.
E) NOTA
What is the 40th term of the arithmetic progression beginning -10, -6, -2, … ?
A) 154
6.
D) 1
A group of 32 calculus students meets for a bridge tournament every Saturday. Before the
game begins, each person in the group shakes hands with every other person exactly once.
How many handshakes occur?
A) 64
5.
B) 610
Find the common ratio of the geometric series
A)
4.
C) 2k
Find the sum of the 20 smallest positive multiples of 3.
A) 1260
3.
B) 2
B) -870
C) -4060
D) -840
E) NOTA
What is the 12th Fibonacci number, F12 ? (Assume Fn  Fn 1  Fn  2 and F1  F2  1 .)
A) 89
B) 144
C) 233
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D) 78
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Sequences and Series Topic Test – Theta Division
9.
James, an eccentric millionaire, has bought a unique 24-story mansion in which every floor
has 5 more rooms than the floor below it. There are 3 rooms on the bottom floor. What is
the total number of rooms in the mansion?
A) 1416
B) 1452
C) 2400
D) 1412
E) NOTA
10. What is the 40th triangular number, T40 ? (Assume T1  1 and Tn  n  Tn 1 .)
A) 1640
B) 780
C) 1600
D) 820
E) NOTA
C) 55
D) -5
E) NOTA
D) 5625
E) NOTA
11. If a n  n 2  2n  7 , what is a 6 ?
A) 31
B) 17
12. Find the sum of the 150 smallest odd positive integers.
A) 22650
B) 22500
C) 11325
13. Find the sum of the terms of the geometric sequence 27, 9, 3, …
A) 40.5
B) 54
C) 40
D) 81
E) NOTA
14. As part of a science experiment, Richard drops a ping-pong ball out of his 3rd story window
(25 m above the ground). From the data, he has determined that the ping-pong ball rebounds
40% of the height from which it falls. How high will it rise on its 6th bounce?
A) 25.6 cm
B) 10.24 cm
C) 64 cm
D) 4.096 cm
E) NOTA
15. What is the 32nd perfect cube? (Assume that 1 is the 1st).
A) 1024
B) 32768
C) 96
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D) 256
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Sequences and Series Topic Test – Theta Division
16. A diagonal is drawn through a square with side length 2 cm. A smaller square is
constructed, one vertex coinciding with the large square and another coinciding with the
midpoint of one side of the large square. The diagonal of the smaller square is drawn,
starting from the coincident vertex of the two squares, where the first diagonal terminated.
(See figure.) If this pattern continues infinitely, what is the length (in cm) of all diagonals
drawn?
B) 4 2  4
A) 4
C) 4 2
D) 4  2 2
E) NOTA
17. Thom and Tom, both avid gamblers, decide to play a game of chance. They take turns
rolling three fair 6-sided dice, continuing forever until one person rolls triples (all three dice
showing the same number). The person who rolls triples is declared the loser. To make
things interesting, Tom agrees to bet $10 that he will win, provided that Thom allows Tom
to roll first. What is the probability that Thom will win the bet?
A)
35
71
B)

18. Evaluate
1
 (2  3i)
n 0
A)
3i  1
10
n
B)
36
71
C)
216
531
D)
215
531
E) NOTA
11  3i
10
D) diverges
E) NOTA
, where i   1 .
1  3i
10
C)
19. Find the product of the twenty smallest positive even numbers.
A) 220  20!
B) 2 20
C) 20!
D) 210  10!
E) NOTA
20. The Fibonacci sequence can be extrapolated before the positive integers such that the 0th
term is 0, the -1st term is 1, and so on. In this extrapolated Fibonacci sequence, what is the
 15 th term?
A) 15
B) -15
C) 610
D) -610
E) NOTA
B) 396900
C) 14910
D) 7770
E) NOTA
35
21. Evaluate:
n
2
n 1
A) 630
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Mu Alpha Theta National Convention: Hawaii, 2005
Sequences and Series Topic Test – Theta Division


22. The formula 2n  1 n 2  n  1 generates the nth centered cube number. What is the sum
of the first 3 centered cube numbers?
A) 35
B) 45
C) 36
D) 87
E) NOTA
23. On Monday, June 6, 2005, Keith paid $10 for lunch. The next day, he bought a bigger lunch
and paid 10% more than before. On each following day, he spent 10% more than the
previous day, until the 10th day, when his lunch cost $10 again. Assuming this pattern
continues indefinitely, how much did he pay for lunch on Monday, June 27, 2005?
A) $13.31
B) $12.10
C) $11
D) $10
E) NOTA
2
24. Find the value(s) of the following continued fraction: 3 
3
2
3
3
A)
3  17
2
B)
13  97
6
C)
3  17
2
D)
.
2
2
3
13  97
6
E) NOTA
25. Each of the five Read sisters picks a real number at random. Amazingly, the numbers they
choose, when ordered from least to greatest, form a geometric progression. The youngest
sister, Brenda, chose the smallest number, 16. The oldest sister, Bryana, chose the largest
number, 36. What is the sum of all five sisters’ chosen numbers?
A) 76
B) 76  20 6
C) 44  40 3
D) 107 
71 6
2
E) NOTA
26. For a given quadratic sequence Q, Q1  10 , Q3  18 , and Q5  27 . What is Q6 ?
A) 37
B) 36
C)
162
5
D)
73
2
E) NOTA
27. Find the value(s) of the common ratio of the geometric sequence a, b, 6, … given that
a  b  1.
A)  3  15
B) 3  15
C) 3  15
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D) 4  15
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Sequences and Series Topic Test – Theta Division
28. If f ( x)  f ( x  1)  2 and f (1)  5 , find f (5) .
A) 5
B) –4
C) –3
D) –1
E) NOTA
21
16
E) NOTA
1 n
.
n
n 0 5

29. Evaluate the sum:
A)
5
16

B)
5
4
C)
25
16
D)
30. The nth Cullen number is Cn  n  2n  1 . Find the number of digits in the base 10
representation of the sum of the first 2005 Cullen numbers (starting with C1  3 ).
A) 607
B) 608
C) 2017
Page 5 of 5
D) 2018
E) NOTA