Chapter 8.2, Problem 26E

To determine

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

Answer to Problem 26E

An expression for the nth term of the arithmetic sequence is $a_n=\frac{13}{2}-\frac{5}{2}n$ .

Explanation of Solution

Given information:

An sequence is given as $4,\frac{3}{2},-1,-\frac{7}{2},\mathrm{...}$ .

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 given sequence.

  $4,\frac{3}{2},-1,-\frac{7}{2},\mathrm{...}$

Consider dbe the common difference and $a_1$ as the first term.

  $\begin{array}{l}{a}_1=4\\ d=a_2-a_1=\frac{3}{2}-4=-\frac{5}{2}\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =4+\left(n-1\right)\left(-\frac{5}{2}\right)\\ =4-\frac{5}{2}n+\frac{5}{2}\\ =\frac{13}{2}-\frac{5}{2}n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=\frac{13}{2}-\frac{5}{2}n$ .