Verify that the given equation holds for n=1, n=2, and n=3 ;...

Verify that the given equation holds for $n=1, n=2,$ and $n=3 ;$ and (b) use mathematical induction to show that the equation holds for all natural numbers.
$$F_{1}+F_{2}+F_{3}+\dots+F_{n}=F_{n+2}-1$$

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Key Concepts

Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with F1 = 1 and F2 = 1. This sequence is fundamental to understanding recurrence relations and serves as a basis for various identities and proofs involving recursive formulas.

Mathematical Induction

Mathematical induction is a proof technique used to demonstrate that a statement holds for all natural numbers. It consists of proving a base case and an inductive step, thereby ensuring that the truth of the statement for one case implies its truth for the subsequent case.

Base Case Verification

Base case verification involves checking that the equation or statement is true for the initial value(s) of the natural numbers. This step provides the foundation upon which the inductive argument is built.

Inductive Step

The inductive step is the part of a proof by induction where one assumes that the statement holds for a particular natural number n (the induction hypothesis) and then proves it for n+1. This step is crucial for extending the validity of the statement from one case to all natural numbers.

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