SEQUENCES

1. A sequence of odd integers begins at 61 and ends at 119. What is the sum of all numbers in this
sequence?
(A) 2610 (B) 2700 (C) 2790 (D) 5400 (E) 6000
2. For every positive integer n that is greater than 1, the function g(n) is defined to be the sum of
all of the odd integers from 1 to n, inclusive. The g(n) could have any of the following units
digits except…?
(A) 1 (B) 2 (C) 4 (D) 6 (E) 9
3. If n is a positive integer and (n+1)(n+3) is odd, then (n+2)(n+4) must be a multiple of which one
of the following?
(A) 3 (B) 5 (C) 6 (D) 8 (E) 16
4. In a sequence, each term starting with the third onward is defined by the following formula
an=an 1 an 2, where n is an integer greater than 2. If the first term of that sequence is 3 and the
second term is 4, what is the value of the 70th of that sequence?
(A) 4 (B) 3 (C) 1 (D) 3 (E) 4
5. If n is a positive integer, what is the maximum possible number of prime numbers in the
following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
6. The sequence of numbers a1, a2, a3, ..., an is defined by an = 1/n - 1/(n+2) for each integer n >=
1. What is the sum of the first 20 terms of this sequence?
(A) (1+1/2) - 1/20
(B) (1+1/2) - (1/21+1/22)
(C) 1 - (1/20 + 1/20)
(D) 1 - 1/22
(E) 1/20 - 1/22
7. In a certain quiz that consists of 10 questions, each question after the first is worth 4 points
more than the preceding question. If the 10 questions on the quiz are worth a total of 360
points, how many points is the third question worth?
(A) 18 (B) 24 (C) 26 (D) 32 (E) 44
8. The set X consists of the following terms: {4,44,444,4444,.....), where the nth term has n 4's in
it for example 10th term is 4444444444. what is the hundreds digit of the sum of the first 45
terms of set S?
(A) 5 (B) 3 (C) 1 (D) 6 (E) 9
9. The sequence a1, a2, …, an, … is such that an=4an–1 –3 for all integers n > 1. If a3=x, then a1=?
(A) 4x–3 (B) 16x–15 (C) (x+3)/4 (D) (x+3)/16 (E) (x+15)/16
10. In a sequence of 12 numbers, each term, except for the first one, is 1211 less than the previous
term. If the greatest term in the sequence is 1212, what is the smallest term in the sequence?
(A) 1211 (B) 0 (C) 1211 (D) 11x1211 (E) 1212
11. The largest number in a series of consecutive even integers is w. If the number of integers is n,
what is the smallest number in terms of w and n?
(A) w– 2n (B) w–n + 1 (C) w– 2(n– 1) (D) n– 6 + w (E) w – n/2
12. In a sequence of 19 numbers, each term except for the first one is greater than the previous
term by 9. If the last term in the sequence is 200, what is the first term in the sequence?
(A) 9 (B) 10 (C) 20 (D) 29 (E) 38
13. If Sn is the sum of the first n terms of a certain sequence and if Sn = n(n+1) for all positive
integers n, what is the third term of the sequence?
(A) 3 (B) 4 (C) 6 (D) 8 (E) 9
14. The "geometric mean" of two numbers a and b is defined as the square root of their product. For
example, the "geometric mean" of 2 and 18 is 36, i.e. 6. In a certain sequence, each term
except for the first two terms is the "geometric mean" of the two previous terms. If the first
term is 2 and the second term is 32, what is the value of the fourth term?
(A) 8 (B) 16 (C) 32 (D) 64 (E) 128
15. In a certain series, each term is m greater than the previous term. If the 17th term is 560 and
the 14th term is 500, what is the first term?
(A) 220 (B) 240 (C) 260 (D) 290 (E) 305
16. An equilateral triangle T2 is formed by joining the mid points of the sides of another equilateral
triangle T1. A third equilateral triangle T3 is formed by joining the mid-points of T2 and this
process is continued indefinitely. If each side of T1 is 40 cm, find the sum of the perimeters of
all the triangles.
(A) 180 cm (B) 220 cm (C) 240 cm (D) 270 cm (E) 300 cm
17. In a certain sequence, each term except for the first term is one less than twice the previous
term. If the first term is 0.75, then the 3rd term is which of the following?
(A) 1.5 (B) 1 (C) 0 (D) 0.5 (E) 2
18. How many three-digit integers are not divisible by 3?
(A) 599 (B) 600 (C) 601 (D) 602 (E) 603
19. There is a sequence such that each term is positive integer and each digit of the terms in the
sequence has 3 to be ordered, what is the value of 100th term?
(A) 126 (B) 192 (C) 232 (D) 252 (E) 342
20. A and B are two multiples of 14, and Q is the set of consecutive integers between A and B,
inclusive. If Q contains 9 multiples of 14, how many multiples of 7 are there in Q?
(A) 20 (B) 19 (C) 18 (D) 17 (E) 16
21. Set S is composed of following real numbers {1/96, 2/96, … , 96/96 }. Two numbers 'a' and 'b' are
selected at random without replacement and are used to calculate result according to equation
2×a×b-a-b+1 and this result is put back into the set S.
This step is done repeatedly until all the numbers in the set are exhausted and set contains only
1 number. What is that last number?
(A) 1/2 (B) 1/6 (C) 1/8 (D) 1/9 (E) 1/12
22. A sequence consists of 16 consecutive even integers written in increasing order. The sum of the
first 8 of these even integers is 424. What is the sum of the last 8 of the even integers?
(A) 488 (B) 540 (C) 552 (D) 568 (E) 584
23. The first term in sequence Q equals 1, and for all positive integers n equal to or greater than 2,
the nth term in sequence Q equals the absolute value of the difference between the nth smallest
positive perfect cube and the (n-1)th smallest positive perfect cube. The sum of the first seven
terms in sequence Q is
(A) 91 (B) 127 (C) 216 (D) 343 (E) 784
24. The sum of four consecutive odd numbers is equal to the sum of 3 consecutive even numbers.
Given that the middle term of the even numbers is greater than 101 and lesser than 200, how
many such sequences can be formed?
(A) 12 (B) 17 (C) 25 (D) 33 (E) 50
25. The sum Sn of the arithmetic sequence a, a+d, a+2d,..., a+(n-1)d is given by Sn = n/2(2a + (n-
1)×d). What is the sum of the integers 1 to 100 inclusive, with the even integers between 25 and
63 omitted?
(A) 4345 (B) 4302 (C) 4258 (D) 4214 (E) 4170
26. The total number of plums that grow during each year on a certain plum tree is equal to the
number of plums that grew during the previous year, less the age of the tree in years (rounded
down to the nearest integer). During its 3rd year, the plum tree grew 50 plums. If this trend
continues, how many plums will it grow during its 6th year?
(A) 43 (B) 41 (C) 38 (D) 35 (E) 32
AL-201 2 Algebra
27. For every positive integer n, the nth term of a sequence is the total of three consecutive
integers starting at n. What is the total of terms 1 through 99 of this series?
(A) 5,250 (B) 10,098 (C) 14,850 (D) 15,147 (E) 15,150
28. A certain club has exactly 5 new members at the end of its first week. Every subsequent week,
each of the previous week's new members (and only these members) brings exactly x new
members into the club. If y is the number of new members brought into the club during the
twelfth week, which of the following could be y?
(A) 51/12 (B) 311 × 511 (C) 312 × 512 (D) 311 × 512 (E) 6012
29. A sequence of numbers (geometric sequence) is given by the expression: g(n)=5 ×(-1/2)n. If the
sequence begins with n = 1, what are the first two terms for which |g(n) - g(n+1)| < 1/1000 ?
(A) g(10), g(11)
(B) g(11), g(12)
(C) g(12), g(13)
(D) g(13), g(14)
(E) g(14), g(15)
30. For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by
(-1)(k+1) × (1/2k). If T is the sum of the first 10 terms in the sequence then T is
(A) Greater than 2
(B) Between 1 and 2
(C) Between 1/2 and 1
(D) Between 1/4 and 1/2
(E) Less than 1/4
31. Find the sum of the first 20 terms of this series which begins this way : -12 + 22 - 32 + 42 - 52 +
62...
(A) 210 (B) 330 (C) 519 (D) 720 (E) 190
32. If the first, third and thirteenth terms of an arithmetic progression are in geometric progression,
and the sum of the fourth and seventh terms of this arithmetic progression is 40, find the first
term of the sequence?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
33. In a certain sequence, every term after the first is determined by multiplying the previous term
by an integer constant greater than 1. If the fifth term of the sequence is less than 1000, what is
the maximum number of non-negative integer values possible for the first term?
(A) 60 (B) 61 (C) 62 (D) 63 (E) 64
34. In a certain sequence of numbers, a1, a2, a3, ..., an, the average (arithmetic mean) of the first
m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?
(A) 100 (B) 55 (C) 21 (D) 19 (E) 1
35. If the infinite sum 1/21 + 1/22 + 1/23 +1/24 +...=1, what is the value of the infinite sum
1/21+2/22+3/23+4/24+....?
(A) 1 (B) 2 (C) 3 (D) (E) Infinite
36. There is a sequence ni such, in which i is a positive integer, ni+1=2ni.
If n1=1, n2=2, n3=4, n4=8, what is the scope including n21?
(A) 100~1,000
(B) 1,000~10,000
(C) 10,000~100,000
(D) 100,000~1,000,000
(E) 1,000,000~
37. Consider a sequence of numbers given by the expression 5 + (n - 1) × 3, where n runs from 1 to
85. How many of these numbers are divisible by 7?
(A) 5 (B) 7 (C) 8 (D) 11 (E) 12

AL-201 3 Algebra
38. Balls of equal size are arranged in rows to form an equilateral triangle. The top most row
consists of one ball, the 2nd row of two balls and so on. If 669 balls are added, then all the balls
can be arranged in the shape of square and each of the sides of the square contain 8 balls less
than the each side of the triangle did. How many balls made up the triangle?
(A) 1540 (B) 2209 (C) 2878 (D) 1210 (E) 1560
39. 2 + 2 + 22 + 23 + 24 + 25 + 26 + 27 + 28 =?
(A) 29 (B) 210 (C) 216 (D) 235 (E) 237
40. 3 + 3 + 3 + 2 × 32 + 2 × 33 + 2 × 34 + 2 × 35 + 2 × 36 + 2 × 37 =
(A) 37 (B) 38 (C) 314 (D) 328 (E) 330
41. 2(22) + 2(23) + 2(24) + ... + 2(43) + 2(44) =
(A) 222(2(23)-1) (B) 222(2(24)-1) (C) 222(2(25)-1) (D) 223(2(21)-1) (E) 223(2(22)-1)
42. What is the value of n if the sum of the consecutive odd integers from 1 to n equals 169?
(A) 47 (B) 25 (C) 37 (D) 33 (E) 29

ANSWER KEY
1. (B) 2. (B) 3. (D) 4. (B) 5. (C) 6. (B) 7. (C) 8. (C) 9. (E) 10. (C)
11. (C) 12. (E) 13. (C) 14. (B) 15. (B) 16. (C) 17. (C) 18. (B) 19. (E) 20. (D)
21. (A) 22. (C) 23. (D) 24. (A) 25. (D) 26. (C) 27. (D) 28. (C) 29. (D) 30. (D)
31. (A) 32. (A) 33. (D) 34. (D) 35. (B) 36. (E) 37. (E) 38. (A) 39. (A) 40. (B)
41. (A) 42. (B)

AL-201 4 Algebra