Chapter 9.4, Problem 68E

To determine

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

  $\begin{array}{l}{a}_1=0\\ a_n=a_{n-1}+2n\end{array}$

Answer to Problem 68E

The sequence have a 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

  $\begin{array}{l}{a}_1=0\\ a_n=a_{n-1}+2n\end{array}$

The Given , $\begin{array}{l}{a}_1=0\\ a_n=a_{n-1}+2n\end{array}$

  $\begin{array}{l}{a}_n=a_{n-1}+2n\\ a_1=0\\ a_2=a_1+2(2)=0+4=4\\ a_3=a_2+2(3)=4+6=10\\ a_4=a_3+2(4)=10+8=18\\ a_5=a_4+2(5)=18+10=28\\ a_6=a_5+2(6)=28+12=40\\ \end{array}$

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

Then

First differences

  $\begin{array}{l}{d}_1=a_2-a_1=4-0=4\\ d_2=a_3-a_2=10-4=6\\ d_3=a_4-a_3=18-10=8\\ d_4=a_5-a_4=28-18=10\\ d_5=a_6-a_5=40-28=12\\ \end{array}$

Second differences

  $\begin{array}{l}{S}_1=d_2-d_1=6-4=2\\ S_2=d_3-d_2=8-6=2\\ S_3=d_4-d_3=10-8=2\\ S_4=d_5-d_4=12-10=2\end{array}$

Since The second differences are equal therefore ,

The sequence have a Quadratic model