Chapter 9, Problem 33RE

Solution

To state: The first six terms and the 12^th term of the sequence:

  $v_1=-3,v_2=1\text{and}{v}_k=v_{k-2}+v_{k-1}\text{for}k\ge 3$

The resultants first six terms are $-3,1,-2,-1,-3,-4$ and the 12^th term is $-76$ .

Given information:

The given sequence is: $v_1=-3,v_2=1\text{and}{v}_k=v_{k-2}+v_{k-1}\text{for}k\ge 3$

Explanation:

Consider the given sequence,

  $v_k=v_{k-2}+v_{k-1}$

The first and second terms are: $v_1=-3,v_2=1$

Now substitute 3 for k for the second term and simplify,

  $\begin{array}{c}{v}_3=v_{3-2}+v_{3-1}\\ =v_1+v_2\\ =-3+1\\ =-2\end{array}$

Now substitute 4 for k for the third term and simplify,

  $\begin{array}{c}{v}_4=v_{4-2}+v_{4-1}\\ =v_2+v_3\\ =1-2\\ =-1\end{array}$

Similarly, find the remaining 3 terms,

  $\begin{array}{c}{v}_5=v_{5-2}+v_{5-1}\\ =v_3+v_4\\ =-2-1\\ =-3\\ v_6=v_{6-2}+v_{6-1}\\ =v_4+v_5\\ =-1-3\\ =-4\end{array}$

The first six terms are $-3,1,-2,-1,-3,-4$ .

Similarly, for finding the 12^th term ninth and tenth terms are required. Then find the other terms first.

  $\begin{array}{c}{v}_7=v_{7-2}+v_{7-1};v_7=v_5+v_6\\ =-3-4\\ =-7\\ v_8=v_{8-2}+v_{8-1};v_8=v_6+v_7\\ =-4-7\\ =-11\\ v_9=v_{9-2}+v_{9-1},v_9=v_7+v_8\\ =-7-11\\ =-18\\ v_{10}=v_{10-2}+v_{10-1};v_{10}=v_8+v_9\\ =-11-18\\ =-29\\ v_{11}=v_{11-2}+v_{11-1};v_{11}=v_9+v_{10}\\ =-18-29\\ =-47\end{array}$

Similarly, the 12^th term is:

  $\begin{array}{c}{v}_{12}=v_{12-2}+v_{12-1}\\ =v_{10}+v_{11}\\ =-29-47\\ =-76\end{array}$

Therefore, the first six terms are $-3,1,-2,-1,-3,-4$ and the 12^th term is $-76$ .