Chapter 7.3, Problem 2Q

Solution

To calculate : The next term in the sequence $-5,10,-15,20,\dots$ and write rule for $n^{th}$ term.

The next term in the sequence $-5,10,-15,20,\dots$ is $-25$ and the rule for $n^{th}$ term is $a_n={\left(-1\right)}^{n}5n$ .

Given information :

The sequence $-5,10,-15,20,\dots$ .

Formula used :

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

Calculation :

Consider the sequence $-5,10,-15,20,\dots$ ,

It can be seen that the consecutive terms have difference of 5 but at every consecutive term has a opposite sign.

So, add 5 to 20 for the next term of the series and put the negative sign.

  $5+20=25$

The next term in series is $-25$ .

The terms can be written as follows.

  $\begin{array}{c}{a}_1=-5\\ ={\left(-1\right)}^1\cdot 5\cdot 1\end{array}$

  $\begin{array}{c}{a}_2=10\\ ={\left(-1\right)}^2\cdot 5\cdot 2\end{array}$

  $\begin{array}{c}{a}_3=-15\\ ={\left(-1\right)}^3\cdot 5\cdot 3\end{array}$

  $\begin{array}{c}{a}_4=20\\ ={\left(-1\right)}^4\cdot 5\cdot 4\end{array}$

In general, the rule for the series can be written as $a_n={\left(-1\right)}^{n}5n$ .

Thus, the next term in the sequence $-5,10,-15,20,\dots$ is $-25$ and the rule for $n^{th}$ term is $a_n={\left(-1\right)}^{n}5n$ .