Chapter 8.2, Problem 85E

To determine

To explain: an expression for the nth term of the arithmetic sequence can be found for given first term and second term of the sequence.

Explanation of Solution

Given information:

An sequence is given with first term $a_1$ and secondterm $a_2$ .

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.

Now, consider the first term and the second term.

  $a_1$ and $a_2$

Consider dbe the common difference.

Now, common difference can be found by subtracting first term from second term.

  $d=a_2-a_1$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =a_1+\left(n-1\right)\left(a_2-a_1\right)\\ =a_1+n{a}_2-n{a}_1-a_2+a_1\\ =2a_1-a_2+n\left(a_2-a_1\right)\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=2a_1-a_2+n\left(a_2-a_1\right)$ .