Answer
The initial conjecture of the Fibonacci sequence is equivalent to ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$.
Work Step by Step
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,
${{a}_{1}}={{a}_{3}}-{{a}_{2}}$
${{a}_{3}}={{a}_{5}}-{{a}_{4}}$
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${{a}_{n-1}}={{a}_{n+1}}-{{a}_{n+2}}$
${{a}_{n}}={{a}_{n+2}}-{{a}_{n+1}}$
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 ${{a}_{1}}+{{a}_{2}}+\cdots +{{a}_{n-1}}+{{a}_{n}}={{a}_{n+2}}-1$.
