Chapter 7.8, Problem 26P

Determine if each sequence is a geometric sequence. If it is, find the common ratio and write the explicit and recursive formulas.

$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$

To determine

To find : If the sequence is a geometric sequence. If it is, find the common ratio and write the explicit and recursive formulas.

Answer to Problem 26P

The given sequence is not geometric.

Explanation of Solution

Given sequence: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\mathrm{.....}$

Calculation:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed , non-zero number called common ratio.

Here,

  $\frac{a_2}{a_1}=\frac{\frac{1}{2}}{1}=\frac{1}{2};\frac{a_3}{a_2}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}$ and $\frac{a_4}{a_3}=\frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}$

That is,

  $\frac{a_2}{a_1}\ne \frac{a_3}{a_2}\ne \frac{a_4}{a_3}$

Since, each term is not found by multiplying the previous term by a fixed constant.

Therefore, the given sequence is not geometric.