Chapter 12, Problem 24RE

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 24RE

The series is neither arithmetic nor geometric.

Explanation of Solution

Given:

The given series is $\frac{3}{2},\frac{5}{4},\frac{7}{6},\frac{9}{8},\frac{11}{10},\mathrm{..}$ .

Calculation:

Consider the given series is,

  $\frac{3}{2},\frac{5}{4},\frac{7}{6},\frac{9}{8},\frac{11}{10},\mathrm{..}$

The difference of the first and the second term is,

  $\frac{5}{4}-\frac{3}{2}=-\frac{1}{4}$

The difference of the third and the second term is,

  $\frac{7}{6}+\frac{5}{4}=-\frac{1}{6}$

Thus, the sequence is not arithmetic.

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

  $\frac{\frac{5}{4}}{\frac{3}{2}}=\frac{5}{6}$

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

  $\frac{\frac{7}{6}}{\frac{5}{4}}=\frac{14}{15}$

Thus, the sequence is not geometric.

Thus, the series is neither arithmetic nor geometric.