Question

In Problems 7–12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.

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

Short answer

The given sequence is neither arithmetic nor geometric.

Step-by-step solution

Step 1. Given information.

Sequence is:$\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},............$

Step 2. To find the difference between successive terms.

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

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

Since difference between successive terms is not constant.

Hence given sequence is not arithmetic sequence.

Step 3. To find the difference between successive terms.

$$\begin{gathered} \frac{3}{4}÷\frac{2}{3}=\frac{3}{4}\times \frac{3}{2} \\ =\frac{9}{8} \end{gathered}$$

$$\begin{gathered} \frac{4}{5}÷\frac{3}{4}=\frac{4}{5}\times \frac{4}{3} \\ =\frac{16}{15} \end{gathered}$$

Since the ratio between two successive term is not constant.

Hence this is not a geometric sequence.