Chapter 11.1, Problem 45PPS

To determine

Determine whether the sequence $3,12,27,48,\mathrm{........}$ is arithmetic, geometric, or neither.

Answer to Problem 45PPS

The sequence is neither geometric nor arithmetic.

Explanation of Solution

Given:

The sequence:$3,12,27,48,\mathrm{........}$

Concept Used:

A sequence is a set of numbers, called terms, arranged in some particular order.

An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference.

A geometric sequence is a sequence with the ratio between two consecutive terms constant.

Calculation:

First Term: $a_1=3;$

Common Difference for Arithmetic sequence:

  $d=\text{2nd}\text{term}-\text{1st}\text{term =}12-3=9\text{or}d=\text{3rd}\text{term}-\text{2nd}\text{term}\text{=}\text{27}-\text{12=15}$

The common difference is not constant, so the sequence is not arithmetic.

Common Ratio for Geometric sequence: $r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{12}{3}=4\text{or}r=\frac{\text{3rd}\text{term}}{\text{2nd}\text{term}}=\frac{27}{12}=\frac{9}{4}$

The common ratios are not constant, so the sequence is not geometric.

Thus, the sequence is neither geometric nor arithmetic.