Chapter 8, Problem 10CT

To determine

Find the recursive rule for the given sequence and the value of $a_9$ .

Answer to Problem 10CT

The recursive rule for the given sequence is $a_n=a_{n-1}-n$ and $a_9=-42$ .

Explanation of Solution

Given:

The given sequence is $2,0,-3,-7,-12,---$ .

Calculation:

The $n\text{th}$ number for the given sequence is

  $\begin{array}{l}2-2=0\\ 0-3=-3\\ -3-4=-7\\ -7-5=-12\end{array}$

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

The recursive equation for the series is $a_n=a_{n-1}-n$ .

  $a_n=a_{n-1}-n$

For the value of $a_9$ .

  $\begin{array}{l}{a}_9=a_{9-1}-n\\ \Rightarrow a_9=a_8-9\\ \left[\begin{array}{l}{a}_5=-12\\ a_6=-12-6=-18\\ a_7=-18-7=-25\\ a_7=-25-8=-33\end{array}\right]\\ \Rightarrow a_9=-33-9\\ \Rightarrow a_9=-42\end{array}$

Hence the recursive rule for the given sequence is $a_n=a_{n-1}-n$ and $a_9=-42$ .