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Homework: Arithmetic and Geometric Sequences

Module by: Kenny M. Felder. E-mail the author

Summary: An updated version of the Homework: Arithmetic and Geometric Sequences module.

Exercise 1

Look at a calendar for this month. Look at the column that represents all the Thursdays in this month.

  • a. What are the dates?
  • b. What kind of sequence do these numbers represent?
  • c. If it is arithmetic, what is dd, the common difference? If geometric, what is rr, the common ratio?
  • d. If that sequence continued, what would be the 100th term?

Exercise 2

How many terms are in the arithmetic sequence 25, 28, 31, 34,...,61?

Exercise 3

Suppose that aa, bb, c c, d d… represents an arithmetic sequence. For each of the sequences below, indicate if it is arithmetic, geometric, or neither.

  • a. a + 2 a+2, b + 2 b+2, c + 2 c+2, d + 2 d+2
  • b. 2 a 2a, 2 b 2b, 2 c 2c, 2 d 2d
  • c. a 2 a 2 , b 2 b 2 , c 2 c 2 , d 2 d 2
  • d. 2 a 2 a , 2 b 2 b , 2 c 2 c , 2 d 2 d

Exercise 4

Find x x to make the sequence

10, 30, 2 x + 8 2x+8

  • a. arithmetic
  • b. geometric

Exercise 5

In class, we showed how the “recursive definition” of an arithmetic sequence t n + 1 = t n + d t n + 1 = t n +d leads to the “explicit definition” t n = t 1 + d ( n - 1 ) t n = t 1 +d(n-1). For a geometric sequence, the recursive definition is t n + 1 = r t n t n + 1 =r t n . What is the explicit definition?

Exercise 6

Suppose a gallon of gas cost $1.00 in January, and goes up by 3% every month throughout the year.

Note:

“Goes up 3%” is the same as “multiplies by 1.03.”
  • a. Find the cost of gas, rounded to the nearest cent, each month of the year. (Use your calculator for this one!)
  • b. Is this sequence arithmetic, geometric, or neither?
  • c. If it keeps going at this rate, how many months will it take to reach $10.00/gallon?
  • d. How about $1000.00/gallon?

Exercise 7

In an arithmetic sequence, each term is the previous term plus a constant. In a geometric sequence, each term is the previous term times a constant. Is it possible to have a sequence which is both arithmetic and geometric?

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