Chapter 8.2, Problem 89E

To determine

To find: Whether it is possible to fill in the blanks from an arithmetic sequence. If yes then find the recursion formula for the sequence.

Answer to Problem 89E

The recursion formula is $3n-10$ and the required sequence is $-7,-4,-1,2,5,8,11$ .

Explanation of Solution

Given:

The first term is $-7$ and the seventh term is $11$ .

Calculation:

Consider the nth term of the sequence is of the form,

  $a_n=a_1+\left(7-1\right)d$

Then, the seventh term of the sequence is,

  $\begin{array}{l}{a}_7=a_1+\left(7-1\right)d\\ 11=-7+6d\\ d=3\end{array}$

Then, the second term of the sequence is,

  $\begin{array}{c}{a}_2=-7+\left(2-1\right)\left(3\right)\\ =-4\end{array}$

The, third term of the sequence is,

  $\begin{array}{c}{a}_3=-7+\left(3-1\right)\left(3\right)\\ =-1\end{array}$

The, fourth term of the sequence is,

  $\begin{array}{c}{a}_4=-7+\left(4-1\right)\left(3\right)\\ =2\end{array}$

The, fifth term of the sequence is,

  $\begin{array}{c}{a}_5=-7+\left(5-1\right)\left(3\right)\\ =5\end{array}$

The sixth term of the sequence is,

  $\begin{array}{c}{a}_6=-7+\left(6-1\right)\left(3\right)\\ =8\end{array}$

The recursion formula is obtained as,

  $\begin{array}{c}{a}_n=-1+\left(n-1\right)\left(3\right)\\ =-7+3n-3\\ =3n-10\end{array}$

Thus, the recursion formula is $a_n=a_1+\left(7-1\right)d$ and the required sequence is $-7,-4,-1,2,5,8,11$ .