Chapter 12, Problem 6CT

To determine

Whether the sequence $6,12,36,144,\mathrm{...}$ is arithmetic, geometric, or neither. If the sequence arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric find the common ratio and sum of the first n terms.

Answer to Problem 6CT

Solution:

The sequence $6,12,36,144,\mathrm{...}$ is neither arithmetic nor geometric.

Explanation of Solution

Given information:

Sequence $6,12,36,144,\mathrm{...}$

Explanation:

Consider the sequence $6,12,36,144,\mathrm{...}$

Check the difference between consecutive terms is constant or not.

$12-6=6$ and $36-12=24$

The difference is not constant. Hence the sequence is not arithmetic.

Now, check if the ratio is common between two consecutive terms.

$\frac{12}{6}=2$ and $\frac{36}{12}=3$

The ratio is not constant. Hence the sequence is not geometric also.

Thus the sequence $6,12,36,144,\mathrm{...}$ is neither arithmetic nor geometric.