Chapter 9, Problem 8PS

(a)

To determine

To prove: that $f_1+f_2+\mathrm{...}+f_n=f_{n+2}-1$ using mathematical induction.

Explanation of Solution

Given information:

Let $f_1,f_2,\mathrm{...},f_n,\mathrm{...}$ be the Fibonacci sequence.

Calculation:

We need to use mathematical induction to prove that $f_1+f_2+\mathrm{...}+f_n=f_{n+2}-1$

Fibonacci sequence is defined as follows

  $\begin{array}{l}{f}_1=1\\ f_2=1\\ f_n=f_{n+2}+f_{n-1}\text{for}n\ge 3\end{array}$

Thus,

  $\begin{array}{c}{f}_3=f_1+f_2\\ =2\\ f_4=f_2+f_3\\ =3\\ f_5=f_3+f_4\\ =5\\ f_6=f_4+f_5\\ =8\end{array}$

For $n=2$

  $\begin{array}{l}{f}_1+f_2=2\text{And}\\ f_4-1=2\end{array}$

Assume

  $\begin{array}{c}{f}_1+f_2+\mathrm{...}+f_k=f_{k+2}-1\\ f_1+f_2+\mathrm{...}+f_k+f_{k+1}=f_{k+2}-1+f_{k+1}\mathrm{...}\text{Substituting}\text{for}{f}_1+f_2+\mathrm{...}+f_k\\ =\left(f_{k+2}+f_{k+1}\right)-1\\ =f_{k+3}-1\\ =f_{\left(k+1\right)+2}-1\end{array}$

Thus, by mathematical induction, the formula is valid for all integers $n\ge 2$ .

(b)

To determine

To find: the sum of the first 20 terms of the Fibonacci sequence.

Answer to Problem 8PS

The Sum of first 20 terms of the Fibonacci sequence is 17710.

Explanation of Solution

Calculation:

From (a) above, we can write the formula for first 20 terms of the sequence, that is,

  $\begin{array}{l}{S}_{20}=f_{22}-1\\ S_{20}=17711-1\\ S_{20}=17710\end{array}$

Thus, the sum of first 20 terms of the Fibonacci sequence is 17710.