Chapter 8, Problem 3Q

To determine

Find the pattern, next term and write a rule for the $n\text{th}$ term of the given sequence.

Answer to Problem 3Q

The rule for the $n\text{th}$ term is $a_n=\frac{n}{10\left(1+n\right)}$ .

The $5\text{th}$ term is $a_5=\frac{5}{60}$ .

Explanation of Solution

Given:

The given sequence is $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50},---$ .

Calculation:

In the given sequence, numerator shows arithmetic sequence $1,2,3,4,---$

  $a_1=1$

  $d=2-1=1$

The rule for the $n\text{th}$ term for numerator is

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ \Rightarrow a_n=1+\left(n-1\right)1\\ \Rightarrow a_n=1+n-1\\ \Rightarrow a_n=n\end{array}$ .

In the given sequence, denominator shows arithmetic sequence $20,30,40,50,---$

  $a_1=20$

  $d=30-20=10$

The rule for the $n\text{th}$ term for numerator is

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ \Rightarrow a_n=20+\left(n-1\right)10\\ \Rightarrow a_n=20+10n-10\\ \Rightarrow a_n=10+10n\\ \Rightarrow a_n=10\left(1+n\right)\end{array}$ .

The rule for the $n\text{th}$ term is $a_n=\frac{n}{10\left(1+n\right)}$ .

Where $n\to$ number of terms.

The $5\text{th}$ term $\left(n=5\right)$ is

  $\begin{array}{l}{a}_5=\frac{5}{10\left(1+5\right)}\\ \Rightarrow a_5=\frac{5}{60}\end{array}$

Hence

The rule for the $n\text{th}$ term is $a_n=\frac{n}{10\left(1+n\right)}$ .

The $5\text{th}$ term is $a_5=\frac{5}{60}$ .