Chapter 12.1, Problem 37E

To determine

To calculate: The $33\text{rd}$ term of the sequence $12.2,10.5,8.8,\dots$

Answer to Problem 37E

The $33\text{rd}$ term of the sequence is $a_{33}=-42.2$ .

Explanation of Solution

Given information:

The sequence $12.2,10.5,8.8,\dots$ .

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

Calculation:

Consider the sequence $12.2,10.5,8.8,\dots$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=10.5-12.2\\ =-1.7\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=8.8-10.5\\ =-1.7\end{array}$

Therefore, common difference is $-1.7$ .

According to question $a_1=12.2$ and $n=33$

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

Apply it, to compute the $33\text{rd}$ term,

  $\begin{array}{c}{a}_{33}=12.2+\left(33-1\right)\left(-1.7\right)\\ =12.2-54.4\\ =-42.2\end{array}$

Thus, the $33\text{rd}$ term of the sequence is $a_{33}=-42.2$ .