Question

Identify the following sequences as arithmetic or geometric. If it is arithmetic state the common difference. If it is geometric state the common ratio -5, 0, 5, 108, 4, 2, 18 16/3, 32/9, 64/2727, 23, 19, 15

Answer 1

We need to identify the type of the sequence and the common difference for arithmetic sequences and common ratio for geometric sequences.

For sequence 1:

`\begin{gathered} \text{Sequence: -5, 0, 5, 10} \\ a_n=a_(n-1)+5 \\ a_n-a_(n-1)=5 \end{gathered}`

The sequence is arithmetic and the common difference is 5.

For sequence 2:

`\begin{gathered} \text{Sequence: 8, 4, 2, 1} \\ a_n=(a_(n-1))/(2) \\ (a_n)/(a_(n-1))=(1)/(2) \end{gathered}`

The sequence is geometric and the common ratio is 1/2.

For sequence 3:

`\begin{gathered} \text{Sequence: }8,\text{ 16/3, 32/9, 64/27} \\ a_n=a_(n-1)\cdot(2)/(3) \\ (a_n)/(a_(n-1))=(2)/(3) \end{gathered}`

The sequence is geometric and the common ratio is 2/3.

For sequence 4:

`\begin{gathered} \text{Sequence: 27, 23, 19, 15} \\ a_n=a_(n-1)-4 \\ a_n-a_(n-1)=-4 \end{gathered}`

The sequence is arithmetic and the common difference is -4.