Chapter 7.3, Problem 3Q

Solution

To calculate : The next term in the sequence $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50}\dots$ and write rule for $n^{th}$ term.

The next term in the sequence $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50}\dots$ is $\frac{5}{60}$ and the rule for $n^{th}$ term is $a_n=\frac{n}{10n+10}$ .

Given information :

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

Formula used :

Look at the consecutive terms and form the required series using the pattern that follows.

Calculation :

Consider the sequence $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50}\dots$ ,

It can be observed that the numerator is getting increased by 1 and denominator is increasing by 10.

The next term of the series is as follows.

  $\frac{4+1}{50+10}=\frac{5}{60}$

The next term in series is $\frac{5}{60}$ .

The terms can be written as follows with denominator as a multiple of 10.

  $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50}\dots =\frac{1}{2\left(10\right)},\frac{2}{3\left(10\right)},\frac{3}{4\left(10\right)},\frac{4}{5\left(10\right)}\dots$

In general, the rule for the series can be written as $a_n=\frac{n}{10n+10}$ .

Thus, the next term in the sequence $\frac{1}{20},\frac{2}{30},\frac{3}{40},\frac{4}{50}\dots$ is $\frac{5}{60}$ and the rule for $n^{th}$ term is $a_n=\frac{n}{10n+10}$ .