# How do you tell whether the sequence 3, 9, 27, 81, 243,... is geometric?

Feb 18, 2017

You look for either a common difference between each pair of terms (arithmetic progression) or a common ratio between each pair (geometric).

#### Explanation:

If you think it might be an arithmetic progression, you must be able to find a common difference between each pair of terms. That does not happen here.

Look at the first few pairs:

$9 - 3 = 6$, but $27 - 9 = 18$ and $81 - 27 = 54$

The differences you calculate in this way are all different. So, this is not an arithmetic progression.

When you calculate the ratio formed by each pair of consecutive terms, you find:

$9 \div 3 = 3$, $27 \div 9 = 3$, and $81 \div 27 = 3$

The same ratio appears every time. This is proof that the progression is geometric.

(You will need this common ratio, often symbolized as $r$, to calculate the $n$th term or the sum of $n$ terms in the series.)