Chapter 9.4, Problem 49E

Solution

To state: The complete table if $F_n$ is the nth term of the Fibonacci sequence and $S_n$ is the nth partial sum of the Fibonacci series which is $S_n={\displaystyle \sum _{k=1}^n{F}_k}$.

  

The resultant complete table is:

n	$F_n$	$S_n$	$F_{n+2}-1$
1	1	1	1
2	1	2	2
3	2	4	4
4	3	7	7
5	5	12	12
6	8	20	20
7	13	33	33
8	21	54	54
9	34	88	88

Given information:

The given nth partial sum of the Fibonacci series is $S_n={\displaystyle \sum _{k=1}^n{F}_k}$.

Explanation:

Fibonacci sequence is a number in which the next number is found out by adding up the previous number.

First find the values of $F_n$ ,

The given values are: $F_1=1\text{and}{F}_2=1$

  $\begin{array}{c}{F}_3=F_1+F_2\\ =1+1\\ =2\\ F_4=F_2+F_3\\ =1+2\\ =3\\ F_5=F_3+F_4\\ =2+3\\ =5\end{array}$

Similarly find the other values of $F_n$ :

  $F_6=8,F_7=13,F_8=21,F_9=34$

Now find the sum $S_n$ ,

  $\begin{array}{c}{S}_1=F_1\\ =1\\ S_2=F_1+F_2\\ =1+1\\ =2\\ S_3=F_1+F_2+F_3\\ =1+1+2\\ =4\\ S_4=F_1+F_2+F_3+F_4\\ =1+1+2+3\\ =7\end{array}$

Similarly find the other values of $S_n$ :

  $S_5=12,S_6=20,S_7=33,S_8=54,S_9=88$

Now find the sum $S_n$ ,

  $\begin{array}{c}{F}_{1+2}-1=F_3-1\\ =2-1\\ =1\\ F_{2+2}-1=F_4-1\\ =3-1\\ =2\\ F_{3+2}-1=F_5-1\\ =5-1\\ =4\end{array}$

Similarly find the other values of $F_{n+2}-1$ :

  $\begin{array}{l}{F}_{4+2}-1=7,F_{5+2}-1=12,F_{6+2}-1=20,\\ F_{7+2}-1=33,F_{8+2}-1=54,F_{9+2}-1=88\end{array}$

The complete table is:

n	$F_n$	$S_n$	$F_{n+2}-1$
1	1	1	1
2	1	2	2
3	2	4	4
4	3	7	7
5	5	12	12
6	8	20	20
7	13	33	33
8	21	54	54
9	34	88	88

It can be observed from the table that the values of $S_n$ and $F_{n+2}-1$ are same. Therefore, the conjecture will be $S_n=F_{n+2}-1$ .