Chapter 8.2, Problem 26E

To determine

To find: an expression for the nth term of the arithmetic sequence for given first term and fifth term.

Answer to Problem 26E

An expression for the nth term of the arithmetic sequence is $a_n=5n-9$ .

Explanation of Solution

Given information:

An sequence is given with first term $a_1=-4$ and fifth term $a_5=16$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the first term and the fifth term.

  $a_1=-4$ and $a_5=16$

Consider dbe the common difference.

Now, expression for fifth term will be

  $\begin{array}{c}{a}_5=a_1+\left(5-1\right)d\\ 16=-4+4d\\ 20=4d\\ d=5\end{array}$

So, the nth term of the arithmetic sequence can be found as

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =-4+\left(n-1\right)\left(5\right)\\ =-4+5n-5\\ =5n-9\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=5n-9$ .