Chapter 9.4, Problem 62E

To determine

To find :The first six terms of the sequence $a_1=2,a_n=a_{n-1}+2$ beginning with the term $a_1$ and the first and second differences of the sequence . Also, state whether the sequence has a perfect linear model, a perfect square model, or neither.

Answer to Problem 62E

The first 6 terms of the sequence are

  $a_1=2,a_2=4,a_3=6,a_4=8,a_5=10,a_6=12$

The first differences are 2,2,2,2,2

The second differences are 0,0,0,0

The sequence has perfectly linear model.

Explanation of Solution

Given:

  $a_1=2,a_n=a_{n-1}+2$

Calculation:

The first term of the sequence is $a_1=2$

The recursive relation for the nth term of the sequence is

  $a_n=a_{n-1}+2$

By substituting $n=2,3,4,5,6$ , we get

  $\begin{array}{l}{a}_2=a_1+2=2+2=4\\ a_3=a_2+2=4+2=6\\ a_4=a_3+2=6+2=8\\ a_5=a_4+2=8+2=10\\ a_6=a_5+2=10+2=12\end{array}$

The first differences of the sequence are

  $\begin{array}{l}{x}_1=a_2-a_1=4-2=2\\ x_2=a_3-a_2=6-4=2\\ x_3=a_4-a_3=8-6=2\\ x_4=a_5-a_4=10-8=2\\ x_5=a_6-a_5=12-10=2\end{array}$

The second differences of the sequence are

  $x_2-x_1=0,x_3-x_2=0,x_4-x_3=0,x_5-x_4=0$

Since, the first difference between any two consecutive terms is constant, therefore, the sequence has perfectly linear model.