Chapter 12.2, Problem 23E

To determine

Calculate the first five terms of the given sequence $a_n=\frac{1}{1+2n}$ and find whether it is arithmetic. If it is arithmetic then also calculate the common difference, and express the $n^{th}$ term of the sequence in the standard form $a_n=a+(n-1)d$ .

Answer to Problem 23E

The first five term of the given sequence $a_n=\frac{1}{1+2n}$ is $\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9}\&\frac{1}{11}$ .

The given sequence $a_n=\frac{1}{1+2n}$ is not in arithmetic.

Explanation of Solution

Given information:

It is given in the question that,

the given sequence is $a_n=\frac{1}{1+2n}$

Formula used:

  $a_n=a+(n-1)d$ & $d\Rightarrow a_n-a_{n-1}=a_{n+1}-a_n$

Where,

  $\begin{array}{l}aisthe{1}^{st}term\\ nisthenumberofterm\\ disthecommondifference\end{array}$

Calculation:

For finding the first five term put $n=1,2,3,4,5$ in the given sequence is $a_n=\frac{1}{1+2n}$

So,

By putting $n=1$ in $a_n=\frac{1}{1+2n}$

  $\begin{array}{l}{a}_1=\frac{1}{1+(2*1)}\\ =\frac{1}{1+(3)}\\ =\frac{1}{4}\end{array}$

By putting $n=2$ in $a_n=\frac{1}{1+2n}$

  $\begin{array}{l}{a}_2=\frac{1}{1+(2*2)}\\ =\frac{1}{1+(4)}\\ =\frac{1}{5}\end{array}$

By putting $n=3$ in $a_n=\frac{1}{1+2n}$

  $\begin{array}{l}{a}_3=\frac{1}{1+(2*3)}\\ =\frac{1}{1+(6)}\\ =\frac{1}{7}\end{array}$

By putting $n=4$ in $a_n=\frac{1}{1+2n}$

  $\begin{array}{l}{a}_4=\frac{1}{1+(2*4)}\\ =\frac{1}{1+(8)}\\ =\frac{1}{9}\end{array}$

By putting $n=5$ in $a_n=\frac{1}{1+2n}$

  $\begin{array}{l}{a}_5=\frac{1}{1+(2*5)}\\ =\frac{1}{1+(10)}\\ =\frac{1}{11}\end{array}$

By using the formula $d\Rightarrow a_n-a_{n-1}=a_{n+1}-a_n$

  $\begin{array}{l}d\Rightarrow a_2-a_1=a_3-a_2\\ \Rightarrow \frac{1}{5}-\frac{1}{3}=\frac{1}{7}-\frac{1}{5}\\ \Rightarrow \frac{3-5}{15}=\frac{5-7}{35}\\ \Rightarrow \frac{-2}{15}=\frac{-2}{35}\\ \Rightarrow \frac{2}{15}=\frac{2}{35}\end{array}$

Here, the sequence not satisfied the given formula concept so the given sequence $a_n=\frac{1}{1+2n}$ which shows that the common difference of two consecutive term is not same.

So , the given sequence $a_n=\frac{1}{1+2n}$ is not in arithmetic.