Sequences and Patterns

1. The 1st term in the geometric sequence below is −13. If it can be determined, what is the 6th term?

Geometric sequences

2021 April(D05) Math Question 1

2011 June(69F) Math Question 2

The 1st term in the geometric sequence below is −9. If it can be determined, what is the 6th term? −9, 18, −36, 72, −144, ...

2023 April(F11) Math Question 3

2019 April(B04) Math Question 5

5. The first term is 1 in the geometric. sequence 1, —3, 9, —27, .... What is the SEVENTH, term of the geometric sequence?

2014 June(72C) Math Question 5

2009 December(68A) Math Question 5

2011 April(67F) Math Question 6

6. Ricardo started a savings account for his daughter Ruth by depositing $500  into the account for her 1st  birthday. For each successive birthday, Ricardo deposits $200  more than the amount deposited for the previous birthday. This is the only money deposited into the account. What is the total amount of money Ricardo will have deposited into the account for Ruth up to and including her 6th birthday?

Arithmetic sequences

2021 December(E23) Math Question 6

2020 July(C01) Math Question 7

7. On the first day of school, Ms. Dubacek gave her third-grade students 6 new spelling words to learn. On each day of school after that, she gave the students 3 new spelling words. How many new spelling words had she given the students by the end of the 21st day of school?

2021 June(D06) Math Question 7

2019 April(B04) Math Question 10

2019 June(72F) Math Question 10

2012 June(70C) Math Question 10

12. The 220 graduating seniors of Madison- High School will sit in the center section of the school auditorium at the graduation ceremony. How many rows of seats will be needed to seat all of the graduating seniors if the first row has 10 seats and each succeeding row has 2 more seats than the previous row?

2018 April(A09) Math Question 12

12. In the standard (x,y) coordinate plane, what is the slope of the line 3x + 8y = 5 ?

2017 December(A10) Math Question 12

12. The table below gives the total charge to rent a moving truck from each of 2 movers for various numbers of miles. For what number of miles would the total charge for renting a moving truck from Ben's be the same as the total charge for renting a moving truck from Ronnie's? (Note: There is a linear relationship between the number of miles and the total charge for both Ben's and Ronnie's.)

2015 April(73G) Math Question 12

2009 June(67C) Math Question 13

2011 April(67F) Math Question 14

2020 July(C01) Math Question 14

15. Mr. Chang made the true statement "At least 1 student in this class is a section leader in the band." Which of the following statements must be FALSE?

2015 June(73C) Math Question 15

What is the sum of the finite geometric series Σ (from n = 1 to 3) of 88/100ⁿ?

2026 April(R4A) Math Question 15

16. The 3 statements below are true for the elements of sets A, B, C, and D.

2018 June(A11) Math Question 16

17. lf it rains in Franklin City on a particular day, the
probability that it will rain there the following day is
0.70, If it does not rain Franklin City on a particular
 day, the probability that it will rain there. the following
day is 0.1. Given that it rained in Franklin City on
Monday, what is the probability that it will NOT rain
in Franklin City on Tuesday of the same week? ·

2019 June(B02) Math Question 17

17. The first 5 terms of a sequence are given in the table below. The sequence is defined by setting al = 9 and a„= an _1+ (n — 1)2 for n ?_ 2. What is the sixth term, a6, of this sequence?

2016 December(74H) Math Question 17

2009 June(67C) Math Question 18

2020 June(C02) Math Question 21

2011 April(67F) Math Question 23

2009 June(67C) Math Question 23

23. The mass of a certain type of bacteria grows exponentially, doubling every 20 minutes. What was the mass, in milligrams, of the bacteria exactly 2 hours after the mass first reached 10 milligrams?

2021 June(D06) Math Question 23

24. Observation of a certain bacteria colony has shown that its population of cells doubles every 3 hours. Given that the initial population of cells in this colony is about 8 million, which of the following values, in millions, would be closest to the number of cells in the bacteria colony after 15 hours?

2015 April(73G) Math Question 24

2009 December(68A) Math Question 25

Consider the repeating decimal 0.7412 (with 12 repeating) in expanded form. What is the 323rd digit to the right of the decimal point?

2025 October(J08) Math Question 25

26. The area of a rectangle is 300 square meters, and its length is 3 times its width. How many meters wide is the rectangle?

2017 December(A10) Math Question 26

26. The 1st and 2nd terms of a certain geometric sequence are 10 and —5, respectively. What is the 5th term of the geometric sequence?

2018 April(A09) Math Question 26

26. In an arithmetic sequence, the 10th term (a10 ) is 30 and the common difference is 2. What is the 1st term (a1 ) ?

2021 December(E23) Math Question 26

2019 June(72F) Math Question 27

27. How many seats are in the 8th row of the auditorium?

2014 December(72G) Math Question 27

31. The 1st term of a geometric sequence is 27, and the 4th term is 64. In terms of n, what is the nth term of the sequence?

2020 December(D03) Math Question 31

2009 April(66F) Math Question 32

2012 April(70G) Math Question 33

2010 April(68G) Math Question 34

34. The sum of a sequence of consecutive odd numbers, where the smallest term is 1, is always a perfect square. For example, 1 + 3 = 22 and 1 + 3 + 5 + 7 = 42- One of the sequences described above has a- sum of 144. What is the largest odd number in the sequence?

2013 June(71C) Math Question 34

2011 June(69F) Math Question 35

2010 December(69A) Math Question 37

2019 December(C03) Math Question 40

The sum of the first 15 positive integers is 120. Which of the following values is the sum of the first 75 positive integers?

2025 October(J08) Math Question 40

41. Beginning next year, Mr. Santana will increase the number of bulbs in the garden each year so that the numbers form a geometric sequence. In 3 years, there will be 162 bulbs in the garden. By what factor will the number of bulbs be multiplied each year?

2016 December(74H) Math Question 41

45. Given today is Tuesday, what day of the week was it 200 days ago?

2017 June(74C) Math Question 45

The solid rectangular prism shown below was built by alternating congruent black cubes and white cubes such that 2 cubes of the same color have at most 1 edge touching. What is the total number of white cubes that were used to build the prism?

2023 April(F11) Math Question 45

46. As x continually increases in value without bound, the

2018 April(A09) Math Question 46

46. Two events are independent if the outcome of one event does not affect the outcome of the other event. One of the following statements does NOT describe independent events. Which one?

2014 December(72G) Math Question 46

2017 December(A10) Math Question 46

47. The recursive formula for a sequence is given below,
where an is the value of the nth term.
a1 = 10
an= a n-1 + 5
Which of the following equations is an explicit
formula for this sequence?

2019 June(B02) Math Question 47

2019 April(B04) Math Question 47

What is the 330th digit after the decimal point in the repeating decimal 4.6238?

2023 April(F11) Math Question 47

2012 June(70C) Math Question 48

48. Consider all positive integers that are multiples of 20 and that are less than or equal to 300. What fraction of those integers are multiples of 15 ?

2016 December(74H) Math Question 48

2019 December(C03) Math Question 49

2010 April(68G) Math Question 49

50. What is the sixth term of the geometric sequence whose second term is —4 and whose fifth term is 32 ?

2015 April(73G) Math Question 50

2013 December(71E) Math Question 50

50. The fraction is equivalent to 0.285714. What is the digit in the 1,001st decimal place of 0.285714 ? (Note: The digit in the 4th decimal place of 0.285714 is 7.)

2016 December(74H) Math Question 50

2012 June(70C) Math Question 50

51. A teacher assigns each of her 18 students a different integer from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?

2015 June(73C) Math Question 51

2021 June(D06) Math Question 52

53. Suppose that equally spaced dots are marked on each side of a regular polygon, with a dot at each vertex, and that the distance between consecutive dots is the same for all sides. The figure below shows 4 equally spaced dots per side, including a dot at each vertex, for an equilateral triangle. Which of the following expressions represents the number of dots for a regular polygon with n equally spaced dots, including one at each vertex, marked on each of its s sides?

2015 April(73G) Math Question 53

2018 December(B05) Math Question 53

53. The 3rd and 4th terms of an arithmetic sequence are 13 and 18, respectively. What is the 50th term of the sequence?

2017 April(74F) Math Question 53

54. A storage facility is currently offering a special rate to customers who sign contracts for 6 months or more. According to this special rate, the first month's rent is $1, and for each month after the first month, customers pay the regular monthly rental rate. The table below shows the storage unit sizes available, the floor dimensions, and the regular monthly rental rate. All the units have the same height. Dania will sign a contract to rent a Size 3 unit for 12 months at the current special rate. The amount Dania will pay for 12 months at the current special rate represents what percent decrease from the regular rental rate for 12 months?

2015 April(73G) Math Question 54

54. The first 4 elements of a pattern are shown below. Each element is composed of small squares that are 18 inches wide and 18 inches long. Each element is a square with both dimensions 18 inches less than the dimensions of the next element. What is the perimeter, in feet, of the 5th element?

Perimeter of Rectangle

Unit Conversions (Feet to inches, miles to feet, hours to seconds)

2020 December(D03) Math Question 54

2019 April(B04) Math Question 55

2019 April(Z15) Math Question 56

2010 April(68G) Math Question 56

2019 April(B04) Math Question 57

57. A finite arithmetic sequence has 7 terms, and the first term is 3/4. What is the difference between the mean and the median of the 7 terms?

2015 June(73C) Math Question 57

2009 June(67C) Math Question 58

2012 December(71A) Math Question 58

59. The sum of the first 30 positive integers is 465. Which of the following is the sum of the first 60 positive integers?

2014 April(71H) Math Question 59

2013 April(71G) Math Question 59

2012 April(70G) Math Question 60

60. Mr. Martin wants to plant 7 trees evenly spaced along a straight fence 300 feet long, with 1 of the trees at each end of the fence. About how many feet apart should he plant the trees?

Sequences and PatternsThere are 82 ACT math questions under Sequences and Patterns

Sub-topics

2021 April(D05) Math Question 1

2011 June(69F) Math Question 2