Chapter 12, Problem 23RE

To determine

To find: Whether the given sequence is arithmetic, geometric or neither. Determine the common difference and the sum of first n terms if the sequence is arithmetic. Determine the common ratio and the sum of First n terms if the sequence is geometric.

Answer to Problem 23RE

The series is neither arithmetic nor geometric.

Explanation of Solution

Given:

The given series is $\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\mathrm{..}$ .

Calculation:

Consider the given series is,

  $\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\mathrm{..}$

The difference of the first and the second term is,

  $\frac{3}{4}-\frac{2}{3}=\frac{1}{12}$

The difference of the third and the second term is,

  $\frac{4}{5}+\frac{3}{4}=\frac{1}{20}$

Thus, the sequence is not arithmetic.

The common ratio of the first and the second term is,

  $\frac{\frac{3}{4}}{\frac{2}{3}}=\frac{9}{8}$

The common ratio of the second and the third term is,

  $\frac{\frac{4}{5}}{\frac{3}{4}}=\frac{16}{15}$

Thus, the sequence is not geometric.

Thus, the series is neither arithmetic nor geometric.