Question

Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form.1,5,25

Answer 1

Geometric sequence:

A gometric sequence has a constant ratio between each pair of consecutive terms.

Arithmatic sequence:

It has a constant difference between each consecutive pair of terms.

For the given sequence,

$1,5,25,\ldots\text{..}$

Check the common difference,

$\begin{gathered} 5-1=4 \\ 25-5=20 \\ \Rightarrow\text{ The common difference is not constant} \end{gathered}$

So, it is not a arithmatic sequence.

Now, check the common ratio,

$\begin{gathered} \frac{5}{1}=5 \\ \frac{25}{5}=5 \end{gathered}$

It implies the given sequence has constant common ratio.

So, the given sequence is geometric.

and commom ratio is 5.