Chapter 11.3, Problem 2ST

(a)

To determine

To calculate:

The formula for $n^{th}$ term for the arithmetic sequence $36,29,22,15,\mathrm{...},$

Answer to Problem 2ST

The formula for $n^{th}$ term for the arithmetic sequence $36,29,22,15,\mathrm{...},$ is $a_n=43-7n$ .

Explanation of Solution

Given information:

  $36,29,22,15,\mathrm{...},$

Formula used:

Common difference: $r=a_n-a_{n-1}$

Arithmetic progression: $a_n=a_1+\left(n-1\right)r$

Calculation:

Consider the sequence as:

  $\begin{array}{l}{a}_1=36\\ a_2=29\\ a_3=22\\ a_4=15\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}r=a_2-a_1=29-36=-7\\ r=a_3-a_2=22-29=-7\\ r=a_4-a_3=15-22=-7\end{array}$

Now, the $n^{th}$ term is calculated as:

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)r\\ a_n=36+\left(n-1\right)\left(-7\right)\\ a_n=36-7n+7\\ a_n=43-7n\end{array}$

(b)

To determine

To calculate:

The value of $t_{19}$ term for the arithmetic sequence $36,29,22,15,\mathrm{...},$

Answer to Problem 2ST

The value of $t_{19}$ term for the arithmetic sequence $36,29,22,15,\mathrm{...},$ is $-90$ .

Explanation of Solution

Given information:

  $36,29,22,15,\mathrm{...},$

Calculation:

The $n^{th}$ term formula from sub-part (a):

  $a_n=43-7n$

Here, $n=19$

  $\begin{array}{l}{a}_{19}=43-7\left(19\right)\\ a_{19}=43-133\\ a_{19}=-90\end{array}$