Chapter 8, Problem 5RE

To determine

To find: an expression for the nth term of the given arithmetic sequence.

Answer to Problem 5RE

An expression for the nth term of the arithmetic sequence is $a_n=\frac{2}{2n-1}$ .

Explanation of Solution

Given information:

An sequence is given as $2,\frac{2}{3},\frac{2}{5},\frac{2}{7},\frac{2}{9},\mathrm{...}$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the given sequence.

  $2,\frac{2}{3},\frac{2}{5},\frac{2}{7},\frac{2}{9},\mathrm{...}=2\left(1,\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9},\mathrm{...}\right)$

Consider dbe the common difference and $a_1$ as the first term.

  $a_1=2\cdot 1-1=1$

Common difference of denominator,

  $d=a_2-a_1=5-3=2$

So, the nth term of the arithmetic sequence can be found as

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =1+\left(n-1\right)\left(2\right)\\ =1+2n-2\\ =2n-1\end{array}$

So, the sequence will have nth term as

  $a_n=\frac{2}{2n-1}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=\frac{2}{2n-1}$ .