Chapter 9.1, Problem 10E

To determine

To Find: The first four terms and the eighth term of a recursively defined sequence.

Answer to Problem 10E

The first $4$ terms and the $8^{th}$ term of the sequence are $-3,2,-1,1$ and $2$ respectively.

Explanation of Solution

Given Information: $v_1=-3$ , $v_2=2$ and $v_n=v_{n-1}+v_{n-2}$ for all $n\ge 3$

Method Used: Substitute a given term in the recursive relation to find its next term.

Calculation: Put $n=3$ in the recursive relation and substitute the given values of $v_1=-3$ and $v_2=2$ to write

  $v_3=v_{3-1}+v_{3-2}=v_2+v_1=2-3=-1$

Similarly, get

  $v_4=v_3+v_2=-1+2=1$

  $v_5=v_4+v_3=1-1=0$

  $v_6=v_5+v_4=0+1=1$

  $v_7=v_6+v_5=1+0=1$

  $v_8=v_7+v_6=1+1=2$

Therefore, the first $4$ terms and the $8^{th}$ term of the sequence are $-3,2,-1,1$ and $2$ respectively.