Chapter 9.4, Problem 67E

To determine

To Check:TheSequence has a perfect linear model , perfect quadratic model or neither

  $\begin{array}{l}{a}_1=2\\ a_n=n-a_{n-1}\end{array}$

Answer to Problem 67E

The sequence neither have a linear model nor Quadratic model

Explanation of Solution

Given:Write the first six terms of the sequence beginning with the given terms . Then calculate the first and the second differences of the sequence . State whether the sequence has a perfect linear model o, perfect quadratic model or neither

  $\begin{array}{l}{a}_1=2\\ a_n=n-a_{n-1}\end{array}$

The Given , $\begin{array}{l}{a}_1=2\\ a_n=n-a_{n-1}\end{array}$

  $\begin{array}{l}{a}_n=n-a_{n-1}\\ a_1=2\\ a_2=2-a_1=2-2=0\\ a_3=3-a_2=3-0=3\\ a_4=4-a_3=4-3=1\\ a_5=5-a_4=5-1=4\\ a_6=6-a_5=6-4=2\\ \end{array}$

Then Let the first difference be denoted by d and the second differences denoted by S

Then

First differences

  $\begin{array}{l}{d}_1=a_2-a_1=0-2=-2\\ d_2=a_3-a_2=3-0=3\\ d_3=a_4-a_3=1-3=-2\\ d_4=a_5-a_4=4-1=3\\ d_5=a_6-a_5=2-4=-2\\ \end{array}$

Second differences

  $\begin{array}{l}{S}_1=d_2-d_1=3-(-2)=5\\ S_2=d_3-d_2=-2-3=-5\\ S_3=d_4-d_3=3-(-2)=5\\ S_4=d_5-d_4=-2-3=-5\\ \end{array}$

Since Neither the First differences nor second differences are equal therefore ,

The sequence does not have a linear model nor Quadratic model