Chapter 9.3, Problem 9E

Solution

To Calculate: The first 4 terms and the 8th term of the recursively defined sequence $c_1=2,c_2\text{=}-1\text{ and }{c}_{k+2}=c_k+c_{k+1}\text{ for }k\ge 1$ .

The first 4 terms and the 8th terms are $2,-1,1,0,\text{ and }3$ respectively.

Given information:

The givens sequence is $c_1=2,c_2\text{=}-1\text{ and }{c}_{k+2}=c_k+c_{k+1}\text{ for }k\ge 1$

Formula used:

Substitute the given values into the given sequence.

Calculation:

Consider the given sequence.

The first term is 2 and second term is $-1$ .

Put $k=1$ into the given sequence.

  $\begin{array}{c}{c}_{1+2}=c_1+c_{1+1}\\ c_3=c_1+c_2\\ =2-1\\ =1\end{array}$

For $k=2$

  $\begin{array}{c}{c}_{2+2}=c_2+c_{2+1}\\ c_4=c_2+c_3\\ =-1+1\\ =0\end{array}$

For $k=3$

  $\begin{array}{c}{c}_{3+2}=c_3+c_{3+1}\\ c_5=c_3+c_4\\ =1+0\\ =1\end{array}$

For $k=4$

  $\begin{array}{c}{c}_{4+2}=c_4+c_{4+1}\\ c_6=c_4+c_5\\ =0+1\\ =1\end{array}$

For $k=5$

  $\begin{array}{c}{c}_{5+2}=c_5+c_{5+1}\\ c_7=c_5+c_6\\ =1+1\\ =2\end{array}$

For $k=6$

  $\begin{array}{c}{c}_{6+2}=c_6+c_{6+1}\\ c_8=c_6+c_7\\ =1+2\\ =3\end{array}$

The first 4 terms and the 8th terms are $2,-1,1,0,\text{ and }3$ respectively.