Chapter 12.2, Problem 57E

To determine

Find whether the following sequence is harmonic.

Answer to Problem 57E

Yes, the term $1,\frac{3}{5},\frac{3}{7},\frac{1}{3},\mathrm{..........}$ is in harmonic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $1,\frac{3}{5},\frac{3}{7},\frac{1}{3},\mathrm{..........}$ .

Concept Used:

In this, use the concept that if the reciprocal of the terms of the sequence form an arithmetic sequence then it is harmonic.

Calculation:

Here, the terms are $1,\frac{3}{5},\frac{3}{7},\frac{1}{3},\mathrm{..........}$

Now, do the reciprocal of the above terms ,

The reciprocal of the terms are $1,\frac{5}{3},\frac{7}{3},3,\mathrm{..........}$ .

Now, do the arithmetic sequence difference,

  $\begin{array}{l}{a}_2-a_1=\frac{5}{3}-1=\frac{5-3}{3}=\frac{2}{3}\\ a_3-a_2=\frac{7}{3}-\frac{5}{3}=\frac{7-5}{3}=\frac{2}{3}\\ a_4-a_3=3-\frac{7}{3}=\frac{9-7}{3}=\frac{2}{3}\end{array}$

From the above , it is seen that all will get the same constant term, so it is in harmonic sequence.

Conclusion:

Yes, the term is in harmonic