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:

$\begin{gathered} \text{Sequence: 8, 4, 2, 1} \\ a_n=\frac{a_{n-1}}{2} \\ \frac{a_n}{a_{n-1}}=\frac{1}{2} \end{gathered}$

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

For sequence 3:

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

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.