## Precalculus (6th Edition) Blitzer

The initial conjecture of the Fibonacci sequence is equivalent to ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$.
The Fibonacci series is $0,1,1,2,3,5,8,13,21,\ldots$ We observe that the next term of the sequence is the sum of its previous two terms, i.e. ${{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}}$ The first two terms are taken as 0 and 1. Though there are many properties of the Fibonacci sequence, only one property is discussed here, which is as follows: Sum of the Fibonacci numbers: The expression for the sum of Fibonacci numbers is ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$ According to the definition of the Fibonacci sequence, ${{a}_{1}}={{a}_{3}}-{{a}_{2}}$ ${{a}_{3}}={{a}_{5}}-{{a}_{4}}$ . . . . ${{a}_{n-1}}={{a}_{n+1}}-{{a}_{n+2}}$ ${{a}_{n}}={{a}_{n+2}}-{{a}_{n+1}}$ After adding these equations, we get: ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-{{a}_{2}}$ Recall ${{a}_{2}}=1$ (from the Fibonacci sequence), ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-{{a}_{2}}$ ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$ Thus, the initial conjecture of the equation is equivalent to ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$.