Chapter 9.4, Problem 61E

To determine

To find :The first six terms of the sequence 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 61E

The first 6 terms of the sequence are

  $a_1=0,a_2=3,a_3=6,a_4=9,a_5=12,a_6=15$

The first differences are 3, 3,3,3,3.

The second differences are 0,0,0,0.

The sequence has perfectly linear model.

Explanation of Solution

Given:

  $a_1=0,a_n=a_{n-1}+3$

Calculation:

The first term of the sequence is $a_1=0$

The recursive relation for the nth term of the sequence is

  $a_n=a_{n-1}+3$

On substitution $n=2,3,4,5,6$ , we get

  $\begin{array}{l}{a}_2=a_1+3=0+3=3\\ a_3=a_2+3=3+3=6\\ a_4=a_3+3=6+3=9\\ a_5=a_4+3=9+3=12\\ a_6=a_5+3=12+3=15\end{array}$

The first differences of the sequence are

  $\begin{array}{l}{x}_1=a_2-a_1=3-0=3\\ x_2=a_3-a_2=6-3=3\\ x_3=a_4-a_3=9-6=3\\ x_4=a_5-a_4=12-9=3\\ x_5=a_6-a_5=15-12=3\end{array}$

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.