Chapter 6.5, Problem 13E

To determine

To find:Whether the sequence $192,24,3,\frac{3}{8},\mathrm{...}$ is arithmetic, geometric or neither.

Answer to Problem 13E

The sequence $192,24,3,\frac{3}{8},\mathrm{...}$ is geometric.

Explanation of Solution

Given information:The sequence $192,24,3,\frac{3}{8},\mathrm{...}$

Calculation:A sequence is said to be arithmetic sequence if every term of the sequence, after the first term, is obtained by adding a constant to the preceding term. The constant is known as common difference.

A sequence is said to be geometric sequence if every term of the sequence, after the first term, is obtained by multiplying a constant to the preceding term. The constant is known as common ratio.

Here, the 1^st term $=192$

Difference between 1^st and 2^nd term $=24-192=-168$

Difference between 2^nd and 3^rd term $=3-24=-21$

Difference between 3^rd and 4^th term $=\frac{3}{8}-3=-\frac{21}{8}$

Since, each term of the sequence is not obtained by adding a constant to the preceding term, the sequence is not arithmetic.

Again, the 1^st term $=192$

Dividing 2^nd term by 1^st term $=24÷192=\frac{1}{8}$

Dividing 3^rd term by 2^nd term $=3÷24=\frac{1}{8}$

Dividing 4^th term by 3^rd term $=\frac{3}{8}÷3=\frac{1}{8}$

And so on.

Since, each term of the sequence is obtained by multiplying a constant $\frac{1}{8}$ to the preceding term, the sequence is geometric.