Chapter 8.2, Problem 19E

To determine

To calculate: the $nth$ term of the sequence and then find $a_{20}$ .

Answer to Problem 19E

The $nth$ term is $-0.8n+3.1$ and $a_{20}=-12.9$ .

Explanation of Solution

Given information:

  $2.3,1.5,0.7,-0.1,\mathrm{........}$ .

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d.

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

Consider ,

  $2.3,1.5,0.7,-0.1,\mathrm{........}$

Now,

  $\begin{array}{l}=a_2-a_1=\left(1.5\right)-(2.3)=-0.8\\ =a_3-a_2=\left(0.7\right)-\left(1.5\right)=-0.8\\ =a_4-a_3=\left(-0.1\right)-\left(0.7\right)=-0.8\end{array}$

Each difference is $(-0.8)$ .

So, the sequence is arithmetic.

The first term of the sequence is $a_1=2.3$ and common difference is $d=(-0.8)$ .

So, the $nth$ term is

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ \text{ =}\left(2.3\right)+(n-1)\left(-0.8\right)\\ \text{ = }2.3-0.8n+0.8\\ \text{ =}-0.8n+3.1\end{array}$

Now, the $20th$ term is $n=20$

Hence, the $nth$ term is $-0.8n+3.1$ and $a_{20}=-12.9$ .