13.1 Homework: arithmetic and geometric sequences

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 $d$ , the common difference? If geometric, what is $r$ , the common ratio?

• D

  If that sequence continued, what would be the 100th term?

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

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

• A

  $a+2$ , $b+2$ , $c+2$ , $d+2$ …

• B

  $2a$ , $2b$ , $2c$ , $2d$ …

• C

  $a^2$ , $b^2$ , $c^2$ , $d^2$ …

• D

  $2^a$ , $2^b$ , $2^c$ , $2^d$ …

Find $x$ to make the sequence

10, 30, $2x+8$

• A

  arithmetic

• B

  geometric

In class, we showed how the “recursive definition” of an arithmetic sequence $t_{n+1}=t_n+d$ leads to the “explicit definition” $t_n=t_1+d(n-1)$ . For a geometric sequence, the recursive definition is $t_{n+1}=r{t}_n$ . What is the explicit definition?

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

• 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?

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?