Chapter 7.3, Problem 6Q

Solution

To calculate : The next term in the sequence $9,36,81,144\dots$ and write rule for $n^{th}$ term.

The next term in the sequence $9,36,81,144\dots$ is $225$ and the rule for $n^{th}$ term is ${\left(3n\right)}^2$ .

Given information :

The sequence $9,36,81,144\dots$ .

Formula used :

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

Calculation :

Consider the sequence $9,36,81,144\dots$ ,

It can be observed that the terms are square of 3,6,9 and 12.

The next term is square of 15.

  ${15}^2=225$

The next term in series is $225$ .

The terms can be written as follows.

  $\begin{array}{c}{a}_1=9\\ ={\left(3\cdot 1\right)}^2\end{array}$

  $\begin{array}{c}{a}_2=36\\ ={\left(3\cdot 2\right)}^2\end{array}$

  $\begin{array}{c}{a}_3=81\\ ={\left(3\cdot 3\right)}^2\end{array}$

  $\begin{array}{c}{a}_4=144\\ ={\left(3\cdot 4\right)}^2\end{array}$

In general, the rule for the series can be written as ${\left(3n\right)}^2$ .

Thus, the next term in the sequence $9,36,81,144\dots$ is $225$ and the rule for $n^{th}$ term is ${\left(3n\right)}^2$ .