Chapter 8.2, Problem 36E

To determine

To find:

Thefirst five terms, common difference and the $n\text{th}$ term of the sequence.

Answer to Problem 36E

Thefirst five terms are $1.5$ , $-1$ , $-3.5$ , $-6$ , $-8.5$ , $d=-2.5$ and the $n\text{th}$ term of an Arithmetic sequence is $4-2.5n$ .

Explanation of Solution

Given information:

  $a_1=1.5$ and $a_{k+1}=a_k-2.5$

To find the first five term.

Substitute $k=1$ in the given formula.

  $\begin{array}{c}{a}_2=a_1+4\\ =1.5-2.5\\ =-1\end{array}$

Substitute $k=2$ in the given formula.

  $\begin{array}{c}{a}_3=a_2+4\\ =-1-2.5\\ =-3.5\end{array}$

Substitute $k=3$ in the given formula.

  $\begin{array}{c}{a}_4=a_3+4\\ =-3.5-2.5\\ =-6\end{array}$

Substitute $k=4$ in the given formula.

  $\begin{array}{c}{a}_5=a_4+4\\ =-6-2.5\\ =-8.5\end{array}$

The first five terms are $1.5$ , $-1$ , $-3.5$ , $-6$ , $-8.5$ .

To find the common difference $d$ subtract any two consecutive terms.

  $\begin{array}{c}d=a_3-a_2\\ =-3.5-\left(-1\right)\\ =-3.5+1\\ =-2.5\end{array}$

The $n\text{th}$ term of an Arithmetic sequence is:

  $a_n=a_1+\left(n-1\right)d$

Substitute $a_1=1.5$ and $d=-2.5$ in the above formula.

  $\begin{array}{c}{a}_n=1.5+\left(n-1\right)\left(-2.5\right)\\ =1.5-2.5\left(n-1\right)\\ =1.5-2.5n+2.5\\ =4-2.5n\end{array}$

Hence, the next five terms are $1.5$ , $-1$ , $-3.5$ , $-6$ , $-8.5$ , $d=-2.5$ and the $n\text{th}$ term of an Arithmetic sequence is $4-2.5n$ .