Chapter 9, Problem 27CLT

(a)

To determine

To calculate: 20^th term

Answer to Problem 27CLT

  $a_{20}=65.4$

Explanation of Solution

Given information: sixth term is $20.6$ and ninth term is $30.2$

Formula needed:n^th term of an arithmetic expression is $a_n=a+\left(n-1\right)d$ where a is first term and d is common difference.

Calculation:Here, $a_6=20.6,\text{and}{a}_9=30.2$

  $\begin{array}{l}\Rightarrow a+\left(6-1\right)d=20.6\\ \Rightarrow a+5d=20.6-----\left(1\right)\\ \\ \Rightarrow a+\left(9-1\right)d=30.2\\ \Rightarrow a+8d=30.2-----\left(2\right)\end{array}$

Subtract equation (1) from (2),

  $\begin{array}{l}\Rightarrow a+8d-\left(a+5d\right)=30.2-20.6\\ \Rightarrow a+8d-a-5d=9.6\\ \Rightarrow 3d=9.6\\ \Rightarrow d=\frac{9.6}{3}\\ \Rightarrow d=3.2\end{array}$

Plugging value of a in equation (1),

  $\begin{array}{l}\Rightarrow a+5\left(3.2\right)=20.6\\ \Rightarrow a+16=20.6\\ \Rightarrow a=20.6-16\\ \Rightarrow a=4.6\end{array}$

Now, the 20^th term is $a_{20}=a+19d$

  $\begin{array}{l}{a}_{20}=4.6+19\left(3.2\right)\\ a_{20}=4.6+60.8\\ a_{20}=65.4\end{array}$

(b)

To determine

To calculate: n^th term

Answer to Problem 27CLT

  $a_n=3.2n+1.4$

Explanation of Solution

Given information: sixth term is $20.6$ and ninth term is $30.2$

Formula needed:n^th term of an arithmetic expression is $a_n=a+\left(n-1\right)d$ where a is first term and d is common difference.

Calculation:Here, $a_6=20.6,\text{and}{a}_9=30.2$

  $\begin{array}{l}\Rightarrow a+\left(6-1\right)d=20.6\\ \Rightarrow a+5d=20.6-----\left(1\right)\\ \\ \Rightarrow a+\left(9-1\right)d=30.2\\ \Rightarrow a+8d=30.2-----\left(2\right)\end{array}$

Subtract equation (1) from (2),

  $\begin{array}{l}\Rightarrow a+8d-\left(a+5d\right)=30.2-20.6\\ \Rightarrow a+8d-a-5d=9.6\\ \Rightarrow 3d=9.6\\ \Rightarrow d=\frac{9.6}{3}\\ \Rightarrow d=3.2\end{array}$

Plugging value of a in equation (1),

  $\begin{array}{l}\Rightarrow a+5\left(3.2\right)=20.6\\ \Rightarrow a+16=20.6\\ \Rightarrow a=20.6-16\\ \Rightarrow a=4.6\end{array}$

Now, the n^th term is

  $\begin{array}{l}{a}_n=4.6+\left(n-1\right)\left(3.2\right)\\ a_n=4.6+3.2n-3.2\\ a_n=3.2n+1.4\end{array}$