Chapter 8.2, Problem 25E

To determine

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

Answer to Problem 25E

An expression for the nth term of the arithmetic sequence is $a_n=\frac{15}{4}-\frac{3}{4}n$ .

Explanation of Solution

Given information:

An sequence is given as $3,\frac{9}{4},\frac{3}{2},\frac{3}{4},0,\mathrm{...}$ .

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.

  $3,\frac{9}{4},\frac{3}{2},\frac{3}{4},0,\mathrm{...}$

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

  $\begin{array}{l}{a}_1=3\\ d=a_2-a_1=\frac{9}{4}-3=-\frac{3}{4}\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =3+\left(n-1\right)\left(-\frac{3}{4}\right)\\ =3-\frac{3}{4}n+\frac{3}{4}\\ =\frac{15}{4}-\frac{3}{4}n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=\frac{15}{4}-\frac{3}{4}n$ .