Chapter 8, Problem 11Q

To determine

Check whether the given sequence is arithmetic, geometric, or neither. Find the rule for the $n\text{th}$ term of the sequence and the value of $a_9$ .

Answer to Problem 11Q

The given sequence is neither arithmetic nor geometric. The $n\text{th}$ term of the sequence is $a_n=\frac{1}{1+n}$ and $a_9=\frac{1}{10}$ .

Explanation of Solution

Given:

The given sequence is $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},---$

Calculation:

In an arithmetic sequence, the difference of consecutive terms, called the common difference, is constant.

Find the common difference between terms in the sequence $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},---$ .

  $\begin{array}{l}\frac{1}{3}-\frac{1}{2}=-\frac{1}{6}\\ \frac{1}{4}-\frac{1}{3}=-\frac{1}{12}\end{array}$

The common difference is not constant.

In a geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratio of any term to the previous term of the sequence $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},---$ .

  $\begin{array}{l}\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}\\ \frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}\end{array}$

The common ratio is not constant.

For the given series, numerator of every term $=1$ and denominator form an arithmetic sequence $2,3,4,5,---$ .

The $n\text{th}$ term of the sequence is

  $a_n=\frac{1}{2+\left(n-1\right)1}=\frac{1}{1+n}$

Substitute the value of $n=9$ in the rule of $n\text{th}$ term.

  $a_9=\frac{1}{1+9}=\frac{1}{10}$

Hence the given sequence is neither arithmetic nor geometric. The $n\text{th}$ term of the sequence is $a_n=\frac{1}{1+n}$ and $a_9=\frac{1}{10}$ .