SAT Practice Math Questions: Arithmetic and Geometric Sequences

By Geraldine Woods, Ron Woldoff

Math sometimes involves recognizing patterns and seeing where those patterns lead. The SAT occasionally asks you to play mathematician with two types of patterns: arithmetic and geometric. The math word for pattern, by the way, is sequence.

Check out this arithmetic sequence: 2, 5, 8, 11, 14. . . . Notice how each number is 3 more than the previous number? In an arithmetic sequence, you always add or subtract the same number to the previous term to get the next term. Another example of an arithmetic sequence is 80, 73, 66, 59. . . . In this one, you’re subtracting 7 from the previous term.

A geometric sequence is similar to an arithmetic sequence, but it works by multiplication or division. In the sequence 2, 6, 18, 54, . . . every term is multiplied by 3 to get the next term. In 88, 44, 22, 11, . . . each term is divided by 2 to get the next term.

The SAT folks often hide these sequences inside a word problem, as in the following practice questions.

Practice questions

  1. The bacteria population in a day-old wad of chewing gum doubles every 3 hours. If there are 100 bacteria at 12:00 noon on Friday, how many bacteria will be present at midnight of the same day?
    • A. 200
    • B. 300
    • C. 800
    • D. 1,600
  2. Author A, an extraordinarily fast writer who zips through a chapter a day, gets paid $100 for her first chapter, $200 for her second, $300 for her third, and so on. Author B, also a member of the chapter-a-day club, gets paid $1 for his first chapter, $2 for his second, $4 for his third, $8 for his fourth, and so on. On the 12th day,
    • A. Author A is paid $76 more.
    • B. Author B is paid $24 more.
    • C. Author A is paid $1,178 more.
    • D. Author B is paid $848 more.

Answers and explanations

  1. D. To solve this problem, make a chart. Because the population doubles every 3 hours, count off 3-hour intervals, doubling as you go:
    • 12:00 (noon) = 100 bacteria
    • 3:00 p.m. = 200 bacteria
    • 6:00 p.m. = 400 bacteria
    • 9:00 p.m. = 800 bacteria
    • 12:00 (midnight) = 1,600 bacteria
    • Or use the formula for 1201
  2. D. Author A’s plan is an arithmetic sequence, increasing by $100 each time, so on the 12th day she’s paid 100 + 11(100) = 100 + 1,100 = $1,200. Author B’s plan is a geometric sequence, multiplied by 2 each time, so on the 12th day, he’s paid

1202

Because $2,048 – $1,200 = $848, author B is paid $848 more.